A NONDEGENERATE MAXIMUM PRINCIPLE FOR THE IMPULSE
CONTROL PROBLEM WITH STATE CONSTRAINTS∗
A. ARUTYUNOV†, D. KARAMZIN‡, AND F. PEREIRA§
Abstract. In this article, a free-time impulsive control problem with state constraints and equality and inequality constraints on the trajectory endpoints is considered. A weakened maximum principle is obtained for problems with data measurable in the time variable, being the time transver-sality conditions deduced with the help of some extra convexity assumption on the state constraints. In the case of smooth problems a nondegenerate maximum principle is derived by using a penalty function method.
Key words. optimal control, impulsive control, maximum principle, state constraints, nonfixed
time, time transversality conditions, nondegeneracy
AMS subject classifications. 49K15, 49N25 DOI. 10.1137/S0363012903430068
1. Statement of the problem. We shall address the following impulse control
optimization problem: J (p, u, µ) = e0(p)→ min, (1.1) dx = f (x, u, t)dt + g(x, t)dµ, t∈ [t0, t1], (1.2) e1(p)≤ 0, e2(p) = 0, (1.3) ϕ(x, t)≤ 0, t ∈ [t0, t1], (1.4)
u(t)∈ U(t) [t0, t1]-a.e., µ≥ 0,
(1.5)
p = (x0, x1, t0, t1), x0= x(t0), x1= x(t1).
Here e1, e2, ϕ are given vector-functions with values in kj, j = 1, 2, 3, respectively, t ∈ 1 is the time variable, µ is a nonnegative scalar valued Borel measure on time
interval [t0, t1], referred to by impulse control, and x is the state variable with values in n. The notation a.e. stands for almost all t ∈ [t0, t1] with respect to Lebesgue
measure. The vector u with values in m is called control. An admissible control
is an essentially bounded measurable function u(t) such that u(t) ∈ U(t) a.e. The vector p∈ 2n+2 is called an endpoint.
∗Received by the editors June 16, 2003; accepted for publication (in revised form) July 19, 2004;
published electronically March 22, 2005.
http://www.siam.org/journals/sicon/43-5/43006.html
†Differential Equations and Functional Analysis Department, Peoples Friendship University of
Russia, Mikluka-Maklai, 6, Moscow 117198, Russia ([email protected]). This author’s research was supported by Russian Foundation for Basic Research grant N02-01-00334.
‡Department of Calculus Mathematics and Cybernetics, Moscow State University, Vorob’yovi
gory, Moscow 119899, Russia (dmitry [email protected]). This author’s research was supported by Russian Foundation for Basic Research grant N02-01-00334.
§Institute for Systems and Robotics, Faculdade de Engenharia, Universidade do Porto, R. Dr.
Roberto Frias, 4200-465 Porto, Portugal (fl[email protected]). This author’s research was supported by INVOTAN and by the Portuguese Science and Technology Foundation under project CorDyAL.
The functions e0, e1, e2, ϕ, g, defining the minimizing functional, endpoint, and
state constraints, and the impulse driven dynamics are continuously differentiable in all their arguments. The vector function f is linear in u, i.e.,
f (t, x, u) = f0(t, x) + F (t, x)u,
(1.6)
continuously differentiable in x for almost all t, and, together with its partial deriva-tives in x, is measurable in t for all fixed (x, u). Both f and its partial derivaderiva-tives in x are bounded on any bounded set and continuous in (x, u) uniformly in t. The set-valued mapping U is measurable in t and bounded, i.e., ∃ c > 0, s.t. |U(t)| ≤ c
∀t. The set U(t) is convex and closed for almost all t.
The triple (p, u, µ) is said to be a control process if ∃ x(t) : x(t0) = x0, x(t1) = x1 and x, u, µ satisfy (1.2). A control process is called admissible if it satisfies all constraints of the considered problem. The admissible process (p∗, u∗, µ∗) is said to be optimal if for any admissible process (p, u, µ), the inequality e0(p∗)≤ e0(p) is true.
The goal for this article is to derive necessary conditions for optimality in the form of a nondegenerate maximum principle for the free-time impulsive control problem (1.1)–(1.5) with control constraints. The maximum principle is said to degenerate if condition (4.3) in our main result, Theorem 4.1, does not hold.
If condition (4.3) does not hold, then it can be easily seen that the maximum con-dition may not be informative in the sense that it holds for any admissible controls (u, µ). Indeed, that is so when ψ(t) = 0 a.e., σr(s) = 0 a.e. (see Theorem 4.1). For
example, we may point to a specific class of problems for which the classical maxi-mum principle always degenerates: the autonomous time-optimal problem with state constraints and fixed endpoints which belong to the boundary of the state constraints. This article pertains to the significant effort made over the years to extend the conventional optimal control theory, addressing systems with trajectories which are continuous, more precisely, absolutely continuous, to systems whose trajectories might present discontinuities.
Despite the already significant body of theory addressing optimal impulsive con-trol problems developed during the last two decades (consider, for example, [2, 4, 5, 7, 13, 10, 14, 15, 16, 17, 18, 20, 21, 22, 23], none of the authors that derived necessary conditions of optimality for impulsive control problems with state constraints, namely, [4, 15, 17], addressed the issue of nondegeneracy of the optimality conditions.
The issue of nondegeneracy is an important one in optimal control and, for con-ventional optimal control problems with state constraints, it was addressed by, among others, [1, 3, 6]. Furthermore, a weakened maximum principle (Theorem 7.1) for free-time impulsive control problems with data merely measurable in the free-time variable was derived, being the time transversality conditions obtained with the help of some extra convexity assumption on the state constraints. This result is also new.
This article is organized as follows. After addressing in the next section the set of hypotheses under which our results are proved, we present and discuss some preliminary concepts. In section 4, we state our main result and present some remarks as well as an illustrative example. Then, some auxiliary results are stated in section 5. Their proof is presented in the appendix. In section 6, we state and prove two results of increasing order of complexity. While in the first result we consider the free-time impulsive optimal control problem without constraints, in the second result only add endpoint state constraints. The weakened version of our main result is stated and proved for the full problem in section 7. Finally, in section 8, we derive our main result.
2. Hypotheses. Let us formulate basic definitions and assumptions.
Definition 2.1. Endpoint constraints are said to be regular if, for any endpoint
vector p = (x0, x1, t0, t1) satisfying (1.3),
(1) the vectors ∂e j
2
∂p(p), j = 1, . . . , k2, are linearly independent;
(2) there exists vector ¯p∈ 2n+2 such that ∂e2 ∂p(p)¯p = 0, ∂ej1 ∂p(p), ¯p > 0 ∀j s.t. ej1(p) = 0.
Definition 2.2. State constraints are said to be regular if, for any (x, t)
satis-fying (1.4), there exists vector q = q(x, t)∈ n such that ϕj
x(x, t), q > 0 ∀j s.t. ϕj(x, t) = 0.
Definition 2.3. Let vector p∗= (x∗0, x∗1, t∗0, t∗1) satisfy endpoint constraints (1.3)
and ϕ(x∗k, t∗k) ≤ 0, k = 0, 1. We say that state constraints are compatible with
endpoint constraints at p∗ if there exists ε > 0 such that {p ∈ 2n+2:|p − p∗| ≤ ε, e1(p)≤ 0, e2(p) = 0} ⊆ {p : ϕ(x
k, tk)≤ 0, k = 0, 1}.
Assumption S. The function f is continuously differentiable in all arguments, and
the set-valued map U (·) is constant. To be more precise, U(t) ≡ U, being U convex and compact.
Definition 2.4. Let Assumption S be in force. The admissible trajectory x(t),
t∈ [t0, t1], is called controllable at the endpoints (with regard to state constraints) if
there exist uk∈ U and mk∈ [0, +∞), k = 0, 1, such that
(−1)k[f(xk, uk, tk) + g(xk, tk)mk, ϕjx(xk, tk) + ϕjt(xk, tk)] < 0 for all j such that ϕj(xk, tk) = 0. Here xk = x(tk), k = 0, 1.
When g ≡ 0 or µ is in class of absolutely continuous measures having density in L∞, then Definition 2.4 becomes a definition of controllability for the associated conventional control optimization problem [1]. A simple argument gives the following sufficient condition for controllability: a trajectory x(·) is controllable at the endpoints if at least one of the two following conditions holds:
• There exists uk∈ U such that ∀j satisfying ϕj(xk, tk) = 0,
(−1)k[f(xk, uk, tk), ϕjx(xk, tk) + ϕjt(xk, tk)] < 0. • (−1)kWj(x
k, tk) < 0 ∀j satisfying ϕj(xk, tk) = 0.
From now on, let Wj(x, t) =ϕj
x(x, t), g(x, t) , j = 1, . . . , k3.
We need Definitions 2.1–2.4 and Assumption S to prove nondegeneracy (Theorem 4.1).
The following convexity assumption is used along with Definition 2.3 to obtain time transversality conditions for problems with data measurable in t and state con-straints (Theorem 7.1).
Assumption C. Let x∗(·) be the reference optimal trajectory on [t∗0, t∗1] and E0∗= (x∗+0 , t∗0), E1∗= (x∗−1 , t∗1). Let the function ϕ be twice continuously differentiable.
There exists δ > 0 such that ˙
Wj(E)≥ 0 ∀E ∈ n× s.t. |E−Ek∗| ≤ δ ∀j, k s.t. ϕj(Ek∗) = Wj(Ek∗) = 0, (2.1)
where ˙Wj =gxg, ϕjx + g, ϕjxxg is the evolution derivative of Wj at any point of
the arc joining the jump endpoints.
Here are a few comments about condition (2.1). It always holds when Wj(Ek∗) = 0. If ϕj(Ek∗) = Wj(Ek∗) = 0, then (2.1) means that function ϕj is convex along jump evolutions in some neighborhood of the endpoint. In this case, examples where (2.1) holds are
• linear systems—g does not depend on x, and ϕ is linear in x; • conventional system—g ≡ 0 (then in both cases ˙Wj≡ 0); and • the function g does not depend on x, and the matrix ϕj
xx is nonnegative
defined.
Finally, (2.1) is true when ˙Wj(E∗ k) > 0.
3. Preliminaries. Let us introduce the adopted notation and present the main
concepts to be used in this article.
Denote by T = [t0, t1] the time interval and by C(T ) the Banach space of
contin-uous functions f : T → 1 with the usual normf
C= maxt∈T|f(t)|. Let V (T ) be
the linear space of functions of bounded variation on T , right continuous on interval (t0, t1), and Vn(T ) be the space of n-vector valued functions x(t) = (x1(t), . . . , xn(t))
such that xj ∈ V (T ), j = 1, . . . , n. Denote by C∗(T ) the topologically dual space to C(T ) (its elements are Borel measures on T ) and by C+∗(T ) the class of nonnegative Borel measures on T .
Given µ∈ C∗(T ), let the distribution function F (t; µ) of the Borel measure µ be defined by
F (t; µ) =
[0,t]
dµ = µ([0, t]), t∈ (0, 1], F (0; µ) = 0.
The variation of µ is an element in C+∗(T ) defined as the Borel measure Var µ =
µ++ µ−, where µ = µ+− µ− is the Jordan decomposition of µ. The total variation of µ is given by the number|µ| = Var µ(T ). The variation of a vector-measure
(vector-function) is defined as the sum of the variation of its components.
We denote by µdand µc, respectively, the discrete and the continuous part of the
measure µ. Let us put Ds(µ) ={r ∈ T : Var µ({r}) > 0}, Cont(µ) = [T \ Ds(µ)] ∪
{t0}∪{t1}. By µ1≤ µ2it is meant that µ2−µ1∈ C+∗(T ). The weak star convergence
of a sequence of measures{µi} to µ is represented by µi w → µ.
Given g ∈ V (T ), let g(s+) = lim
t→s, t>sg(t), g(s−) = limt→s, t<sg(t) be,
re-spectively, the right and the left limits of g at point s. Function g defines the Borel measure ν[g] by the formula
ν[g]([0, t]) = g(t+)− g(0), t ∈ [0, 1].
Thus, according to our notation, ν[F (t; µ)] = µ. The measure corresponding to the length is denoted by L, L = ν[t]. If x ∈ Vn(T ), then ν[x] is a vector measure with
components ν[xj], j = 1, . . . , n. Var|b
a[g] denotes the variation of the measure ν[g] on
[a, b].
We present the concept of the solution to (1.2), following the one given first in [8, 9] and later completed and further discussed in [2, 13, 11, 12, 20, 23, 5, 16, 19]. Let
us consider the function ξ(x, r, ∆) :n×1×1→ n, with ξ(x, r, ∆) = α
r(1), being αr(s) = αr(s; x, ∆) the solution to the following conventional differential equation:
˙
αr= g(αr, r)∆, s∈ [0, 1], αr(0) = x.
Definition 3.1. The function x(t) is called the solution to (1.2) if x(t0) = x0
and, ∀t > t0, x(t) = x0+ t t0 f (x, u, s)ds+ [t0,t] g(x, s)dµc+ r∈Ds(µ), r≤t [ξ(x(r−), r, µ({r}))−x(r−)].
Here, µ({r}) is the atom of µ at point r.
Let φ(x, t) be a scalar continuous function onn× 1 and consider the following
concept of graph completion supremum: gc sup
t∈T
φ(x, t) = max t∈T smax∈[0,1]
φ(αt(s; x(t−), µ({t})), t).
Now, we are in position to precise the definition of trajectory admissability introduced in section 1.
Definition 3.2. Inequality (1.4), i.e., ϕ(x, t) ≤ 0, is understood in the sense
that for j = 1, . . . , k3, gc sup ϕj(x, t)≤ 0. This means that the trajectory x(t) satisfies the state constraints (1.4) if and only if
(1) ϕ(x(t), t)≤ 0 ∀t ∈ [t0, t1], and
(2) ϕ(αr(s), r)≤ 0 ∀s ∈ [0, 1] and ∀r ∈ Ds(µ).
4. The main result. In this section, we state nondegenerate necessary
condi-tions for optimal control problems with state constraints and equality and inequality endpoint constraints derived in its full generality.
Let λ = (λ0, λ1, λ2) and consider the following scalar functions:
H(x, u, ψ, t) =f(x, u, t), ψ , Q(x, ψ, t) =g(x, t), ψ , l(p, λ) = 2 j=0 ej(p), λj .
Theorem 4.1. Let (p∗, u∗, µ∗) be a solution to problem (1.1)–(1.5). Suppose
that Assumption S is in force, the state constraints are compatible with endpoint con-straints, the state and endpoint constraints are regular, and the optimal trajectory is controllable.
Then, there exist
• number λ0≥ 0, vectors λ1∈ k1, λ1≥ 0, λ2∈ k2, • vector function ψ ∈ Vn(T∗), • scalar function φ ∈ V (T∗), • vector measure η = (η1, . . . , ηk3), ηj∈ C∗ +(T∗), s.t. Ds(µ∗)∩ Ds(ηj) =∅ ∀j, and
• for every atom r ∈ Ds(µ∗), there exist its own – vector function σr∈ Vn([0, 1]),
– scalar function θr∈ V ([0, 1]), and
such that, for all t∈ (t∗0, t∗1] and all s∈ [0, 1], ψ(t) = ψ0− t t∗0 Hx(s)ds− [t∗0,t] Qx(s)dµ∗c+ [t∗0,t] ϕx(x∗, s)dη + Σ(ψ, t), Σ(ψ, t) = r∈Ds(µ∗), r≤t [σr(1)− ψ(r−)], φ(t) = φ0+ t t∗0 Ht(s)ds + [t∗0,t] Qt(s)dµ∗c − [t∗0,t] ϕt(x∗, s)dη + Θ(φ, t), (4.1) Θ(φ, t) = r∈Ds(µ∗), r≤t [θr(1)− φ(r−)], ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ dα∗r(s) = g(α∗r(s), r)∆∗rds, dσr(s) =−gx(α∗r(s), r)σr(s)∆∗rds + ϕx(α∗r(s), r)dηr, dθr(s) =gt(αr∗(s), r), σr(s) ∆∗rds− ϕt(αr∗(s), r)dηr, α∗r(0) = x∗(r−), σr(0) = ψ(r−), θr(0) = φ(r−), ∆∗r= µ∗({r}), ψ0= ∂l ∂x0 (p∗, λ), ψ1=− ∂l ∂x1 (p∗, λ), φ0=−∂l ∂t0 (p∗, λ), φ1= ∂l ∂t1 (p∗, λ), g(α∗ r(s), r), σr(s) = 0 ∀r ∈ Ds(µ∗), supp(ηjr)⊆ {s ∈ [0, 1] : ϕj(α∗r(s), r) = Wj(α∗r(s), r) = 0} ∀j, λ1, e1(p∗) = 0, ϕj(x∗(t), t) = 0 ηj-a.e. ∀j, max
u∈UH(u, t) = H(t) a.e., maxu∈UH(u, t) = φ(t) ∀t ∈ (t ∗ 0, t∗1), (4.2) Q(t)≤ 0 ∀t, Q(t) = 0 µ∗-a.e., λ0+L({t : |ψ(t)| > 0}) + r∈Ds(µ∗) L({s : |σr(s)| > 0})∆∗r= 1. (4.3)
In this result and from now on, we adopt the following short notation: T∗ = [t∗0, t∗1], ψk = ψ(t∗k), φk= φ(t∗k), ∆∗k= µ∗({t∗k}), k = 0, 1, H(t, u) = H(x∗(t), u, ψ(t), t), H(t) = H(t, u∗(t)), Q(t) = Q(x∗(t), ψ(t), t), Hx(t) = Hx(x∗(t), u∗(t), ψ(t), t), . . . , etc.
In other words, if H, Q, or their partial derivatives miss some of arguments x, ψ, u, then it is understood that the values x∗(t), ψ(t), and u∗(t) are considered in their place.
The proof is presented in sections 6, 7, and 8. It is based on a penalty function method [1], results from [16, 20], and on the reduction to the so-called v-problem (Proposition 5.7).
Remark 1. From Theorem 4.1, we deduce Var ν[Q] ≤ const(Var η + L). This
implies that the function Q(t) is continuous on Ds(µ∗).
Remark 2. The time transversality conditions in Theorem 4.1 can be rewritten
as follows: max u∈U H(β ∗ k, u, γk, t∗k) + (−1) k+1 ∆∗k 1 0 gt(α∗k(s), t∗k), σk(s) ds − [0,1] ϕt(α∗k(s), t∗k)dηk− ϕt(x∗k, t∗k)η({t∗k}) (4.4) −∂l ∂tk (p∗, λ) = 0, k = 0, 1.
From now on, β0∗ = x∗(t∗+0 ), β1∗ = x∗(t∗−1 ), γ0∗ = ψ(t∗+0 ), γ1∗ = ψ(t∗−1 ), ∆∗k =
µ∗({t∗k}), k = 0, 1, being αk∗, σk, and ηk, the jump evolution elements at point t∗k, k = 0, 1.
Below, we give a simple example showing that the maximum principle proved in Theorem 4.1 may degenerate when the controllability condition (Definition 2.4) does not hold.
Example 4.2. Consider optimal problem
dx = 2tdµ, t∈ [0, 1]; x0= 0, x1= 1; x≥ t2,
[0,1]
1dµ→ min .
Now, we show that minimum is reached for the Lebesgue measure µ∗ = L. The
corresponding optimal trajectory is the parabola x∗(t) = t2. In fact, this statement is
a consequence of the inequality F (t; µ)≥ t ∀t ∈ [0, 1], which holds for any admissible measure µ. Let us prove it. We have
x(t)≥ t2 ⇒
[0,t]
2tdµ≥ t2.
By integrating by parts, we have 2F (t; µ)t≥ t2+ 2 [0,t]F (s; µ)ds and, thus, F (t; µ)≥ t 2+ 1 t [0,t] F (s; µ)ds.
It follows that F (t; µ) ≥ t/2. By substituting in the right part of the obtained inequality, we arrive at the more precise estimate F (t; µ) ≥ 3t/4. By repeating the procedure, we get, for step n, F (t; µ)≥ (2n− 1)t/2n, and, by passing to the limit as n→ ∞, we establish F (t; µ) ≥ t. Thus µ∗=L.
However, the maximum principle proved in Theorem 4.1 degenerates for this problem. Indeed, the impulsive maximum condition yields 2tψ(t)− λ0 = 0 for a.e.
t ∈ [0, 1]. By passing to the limit when t → 0 and by bearing in mind that ψ(t) is
bounded, we conclude that λ0= 0. Hence, ψ(t) = 0∀t ∈ (0, 1). It follows that (4.3)
does not hold. This happens because the optimal trajectory is not controllable.
5. Lemmas and propositions. In this section, we compile a set of auxiliary
lemmas and propositions that we will use in the proof of the main result. Their proofs appear in the appendix. While Lemmas 5.1 to 5.5 are of independent interest in themselves, Propositions 5.6 and 5.7 support the progressive reduction strategy adopted in the proof of Theorems 4.1, 6.1, 6.2, and 7.1.
Lemma 5.1. Given a sequence of measures µi∈ C+∗(T ), µi→ µ, and a sequencew
of functions fi∈ C(T ), such that
(1) there exists a sequence of absolutely continuous measures ηi ∈ C+∗(T ), ηi w → η such that Ds(µ)∩ Ds(η) = ∅ and Var ν[fi]≤ cηi ∀i;
(2) fi(t)→ f(t) ∀t ∈ Cont(η), where f ∈ V (T ). Then [t0,t1] fi(t)dµi→ [t0,t1] f (t)dµ.
Remark 3. The following example shows that we cannot omit requirement (1) in
Example 5.2. Let T = [0, 1], and dµi= δti(t) is the sequence of Dirac’s measures, concentrated at points ti= 1i. Consider the following sequence of (bump) functions {fi}: fi(t) = ⎧ ⎨ ⎩ it, t∈ [0, i−1], 2− it, t ∈ [i−1, 2i−1], 0, t∈ [2i−1, 1]. It is clear that µi w
→ µ, where dµ = δ0(t), and fi(t) → 0 ∀t ∈ [0, 1]. Nevertheless
[0,1]fi(t)dµi = 1∀i, while
[0,1]f (t)dµ = 0. The fact is that for the sequences {µi}
and{fi} constructed above, there is no sequence {ηi} satisfying the requirements of
Lemma 5.1. Indeed, Var ν[fi] w
→ η with dη = 2δ0(t), but Ds(µ)∩ Ds(η) = {0} = ∅. Lemma 5.3. Let all the hypotheses of Lemma 5.1 hold. Let xi(t0) = x(t0) = 0, i∈ N, and ∀t ∈ (t0, t1], xi(t) = [t0,t] fi(s)dµi, x(t) = [t0,t] f (s)dµ.
For any given convergent sequence {ti}, ti → s ∈ T , where s /∈ Ds(µ), we have that xi(ti)→ x(s).
Lemma 5.4. Let µi ∈ C+∗(T ), µi → µ, uw i → u weakly in L2w (T ), ui(t) ∈ U(t)
a.e., x0,i→ x0, and given a sequence of vector-functions xi∈ Vn(T ) such that dxi= f (xi, ui, t)dt + g(xi, t)dµi, t∈ T, xi(t0) = x0,i.
If gc supt∈T|xi(t)| ≤ const ∀i, then xi(t) → x(t) ∀t ∈ Cont(µ), where x(t) is the solution to (1.2). Furthermore, gc supt∈T|xi(t)| → gc supt∈T|x(t)|.
Lemma 5.5. Let the sequences{fi} and {ηi} be such that fi∈ C(T ), ηi∈ C+∗(T ),
and fi → f ∈ C(T ) uniformly and ηi w
→ η ∈ C∗
+(T ), respectively. Assume also that
supp(η)⊂ W0={t : f(t) = 0}. Consider the sequence of functions {xi} defined by xi(t) =
[t0,t]
fi(s)dηi, t > t0, xi(t0) = 0.
If the function f is Lipschitz continuous, then Var|T[xi]→ 0.
Reduction R1. In parallel with initial problem (1.1)–(1.4), denoted (P ), we shall consider problem (P1): (P1) : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ e0(p)→ min, dx = f (x, u, t)dt + g(x, t)dµ, dαk= g(αk, θk)dυk, αk(t1−k) = xk, dθk = 0, θk= tk, k = 0, 1, t∈ [t0, t1], e1(p)≤ 0, e2(p) = 0, ϕ(x, t)≤ 0, ϕ(αk, θk)≤ 0, k = 0, 1, u(t)∈ U(t) a.e., µ, υk≥ 0, k = 0, 1,
p = (ξ0, ξ1, t0, t1), xk = x(tk), ξk= αk(tk), k = 0, 1.
Proposition 5.6. Problems (P ) and (P1) are equivalent. This means that for
ev-ery admissible process (p, u, µ) of (P ), there exists an admissible process (˜p, ˜u, ˜µ, υ0, υ1) of (P1) such that e0(p) = e0(˜p), and, conversely, for each admissible process of (P1), there exists one of (P ) yielding the same cost.
Reduction R1permits us to focus on simpler problems in which the control
mea-sure has no atoms at the initial and the final points of the time interval.
Remark 4. All the results obtained in this paper for problem (P ) can easily be
written to problem (P1) because the systems corresponding to measures µ, υ0, and υ1 are dynamically independent.
Reduction R2. This is a reduction to the so-called v-problem [1]. Let Assumption
S be in force. Consider problem (P2) (v-problem).
(P2) : ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ e0(p)→ min, dx = (v + 1)f (x, u, χ)dt + g(x, χ)dµ, t∈ [t0, t1], dχ = (v + 1)dt, e1(p)≤ 0, e2(p) = 0, p = (x0, x1, χ0, χ1), ϕ(x, χ)≤ 0, u(t)∈ U, |v(t)| ≤ 1/2 a.e., µ ≥ 0.
Proposition 5.7. Problems (P ) and (P2) are equivalent.
6. Primary problems. We start by considering the optimal control problem
without state and endpoint constraints, i.e., (1.1)–(1.2) and (1.5), for which a max-imum principle is given in Theorem 6.1. Then, the complexity of the problem is increased by adding endpoint constraints (1.3), and the necessary conditions of opti-mality presented in Theorem 6.2 are derived. State constraints will not be considered in this section.
Theorem 6.1. Let (p∗, u∗, µ∗) be an optimal process for the problem specified by (1.1), (1.2) and (1.5). Then, there exists a function ψ∈ Vn(T∗) such that
dx∗= Hψ(t)dt + Qψ(t)dµ∗, dψ =−Hx(t)dt− Qx(t)dµ∗, t∈ T∗, (6.1) ψ0= ∂e0 ∂x0 (p∗), ψ1=− ∂e0 ∂x1 (p∗), (6.2) max
u∈U(t)H(u, t) = H(t) a.e.,
(6.3) Q(t)≤ 0 ∀t, Q(t) = 0 ∀t ∈ supp(µ∗), (6.4) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ess lim sup
t→t∗k max u∈U(t)H(β ∗ k, u, γk, t) +(−1)k+1 ∆∗k 1 0 gt(α∗k, t∗k), σk ds − ∂e0 ∂tk (p∗) ≥ 0,
ess lim inf
t→t∗k umax∈U(t)H(β ∗ k, u, γk, t) +(−1)k+1 ∆∗k 1 0 gt(α∗k, t∗k), σk ds − ∂e0 ∂tk (p∗) ≤ 0, k = 0, 1, (6.5)
ess lim inf
t→t∗k umax∈U(t)H(x ∗ k, u, ψk, t) + (−1)k ∂e0 ∂tk (p∗)≤ 0. (6.6) Here, β0∗ = x∗(t∗+0 ), β∗1 = x∗(t∗−1 ), γ0 = ψ(t∗+0 ), γ1 = ψ(t∗−1 ), ∆∗k = µ∗({t∗k}), k = 0, 1, are the control atoms at the time endpoints (see last remark in section 4), and the pair (α∗k, σk), k = 0, 1, in (6.5) corresponds to the solution (6.1) in atom t∗k. According to our solution concept, this means that the pair (α∗k, σk) satisfies
˙ α∗k = g(α∗k, t∗k)∆k∗, α∗k(0) = x∗(t∗−k ), ˙σk =−gx(α∗k, t∗k)σk∆∗k, σk(0) = ψ(t∗−k ), s∈ [0, 1], k = 0, 1. (6.7)
Proof. The proof of (6.1)–(6.4) is given in [16, 20]. So we only have to show that
the time-transversality conditions (6.5) hold. We shall do it using the scheme from [1].
We start by assuming that measure µ∗ has no atoms at points t∗k, i.e., ∆∗k = 0,
k = 0, 1. Let us prove (6.5) when k = 1.
Fix ε > 0 such that t∗1−ε ∈ Cont(µ∗) and let Tε= [t∗1−ε, t∗1]. Define the measure µε:= µ∗+ µ∗(Tε)δ(t∗1− ε) on [t∗0, t∗1− ε]. Let pε = (x∗0, xε(t∗1− ε), t∗0, t∗1− ε), where xε is the trajectory associated to (x∗0, u∗, µε). For ε sufficiently small and from the
optimality of the process (p∗, u∗, µ∗), i.e.,
e0(pε)− e0(p∗)≥ 0 ∀ε > 0, we have that ∂e0 ∂x1 (p∗), ∆xε − ε∂e0 ∂t1 (p∗) + o(|∆xε|) + o(ε) ≥ 0, (6.8) where ∆xε= xε(t∗1− ε) − x∗1.
Let ξ∗ε = ξ(x∗(t∗1− ε), t∗1− ε, µ∗(Tε)). Then, by definition ∆xε = ∆x1ε+ ∆x2ε,
where ∆x1ε=− Tε f (x∗, u∗, t)dt, ∆x2ε= ξε∗− x∗(t∗1− ε) + Tε g(x∗, t)dµ∗ .
The estimate |∆x1ε| ≤ const ε is obvious. Let us show that |∆x2ε| ≤ const εµ∗(Tε).
Since|∆x2
ε| = 0 whenever µ∗(Tε) = 0, we assume that µ∗(Tε) > 0∀ε > 0.
Let us construct a sequence of absolutely continuous measures{µi} having density
˙
µi= mi> 0 a.e., defined on the segment Tεsuch that µi w
→ µ∗weakly* on Tε, µi(Tε) = µ∗(Tε). Let xi be a continuation of the solution x∗ on segment Tε, corresponding to
the measure µi (such a continuation exists when i is sufficient large). By Lemma 5.4, xi(t)→ x∗(t) ∀t ∈ Cont(µ∗). Then, ∆xε,i = xε(t∗1− ε) − xi(t∗1)→ ∆xε as i → ∞.
Furthermore, ∆x1
ε,i→ ∆x1ε and ∆x2ε,i→ ∆x2ε, being
∆x1ε,i=− Tε f (xi, u∗, t)dt, ∆x2ε,i= ξ∗ε− x∗(t∗1− ε) + Tε g(xi, t)midt . To estimate|∆x2
ε,i|, let us consider the equation
˙
x2,i= g(x1,i+ x2,i, t)mi, t∈ Tε, x2,i(t∗1− ε) = x∗(t∗1− ε).
(6.9)
Here, x1,i(t) =
t
t∗1−εf (xi, u∗, τ )dτ and xi= x1,i+ x2,i.
Define the functions πi: Tε→ [0, 1] as follows: πi(t) =
F (t; µi)− F (t∗1− ε; µi) µ∗(Tε)
, t∈ Tε.
Obviously, πi is absolutely continuous and dπdti > 0 a.e. Therefore, there exists the
inverse function π−1i : [0, 1] → Tε which is also strictly monotone and absolutely
continuous. By the change of variable s = πi(t) in (6.9), and by putting αi(s) = x2,i(πi−1(s)), we arrive at the equation
˙
By definition ξε∗= α(1), where α(s) satisfies ˙ α = g(α, t∗1− ε)µ∗(Tε), s∈ [0, 1], α(0) = x∗(t∗1− ε). From here,|αi(s)− α(s)| ≤ µ∗(Tε) s 0 c(|αi(τ )− α(τ)| + Tε|f(xi, u ∗, θ)|dθ + ε)dτ ∀s.
From Gronwall’s inequality,|∆x2ε,i| = |αi(1)− α(1)| ≤ const εµ∗(Tε) and,
there-fore,
|∆x2
ε| ≤ const εµ∗(Tε) = o(ε).
The last equality holds from the fact that µ∗(Tε)→ 0 as ε → 0.
In view of the obtained estimates and the already proved transversality conditions (6.2), inequality (6.8) becomes ψ1, Tε f (x∗, u∗, t)dt − ε∂e0 ∂t1(p ∗) + o(ε)≥ 0.
From here, we may write t∗1 t∗1−ε max u∈U(t)H(u, ψ1, t)dt− ε ∂e0 ∂t1 (p∗) + o(ε)≥ 0,
and, by dividing the last inequality by ε > 0 and taking the limit as ε goes to zero, we obtain
ess lim sup
t→t∗1 max u∈U(t) H(x∗1, u, ψ1, t)− ∂e0 ∂t1 (p∗)≥ 0.
This corresponds to the first inequality in (6.5).
Now, we consider the other inequality for the final endpoint. Let us put µ∗([t∗1, +∞))
= 0. The control function u∗ is continued to the right beyond t∗1as follows:
u∗(t)∈ Arg max
u∈U(t)
H(x∗1, u, ψ1, t), t > t∗1.
The trajectory x∗is continued as a solution to (1.2), corresponding to the constructed
u∗, on the segment [t∗1, t∗1+ δ] for a sufficient small δ > 0. For ε ∈ (0, δ], let pε =
(x∗0, x∗(t∗1+ ε), t∗0, t∗1+ ε). By using the same arguments as above, we arrive at the
second inequality in (6.5) for k = 1. Similar arguments allow us to show these inequalities for k = 0.
Now, let us show that conditions (6.5) hold when ∆∗k > 0, k = 0, 1. For this
purpose let us consider the problem (P1) as a reduction (via R1) of (P ).1
If the process (p∗, u∗, µ∗) is a solution to (P ), then the process (β0∗, β1∗, t∗0, t∗1, u∗,
∆∗0, ∆∗1, ˜µ) is a solution to (P1). Here, ˜µ = µ∗−k=01 ∆∗kδt∗k(t), and β0∗, and β1∗ are
as defined above. The measure ˜µ has no atoms in t∗k, k = 0, 1, and the maximum principle for this problem has just been proved.
1Here, (P ) and (P
1) are considered as in section 5 but without constraints. To be more precise,
if (P ) is a problem without constraints, then (P1) is defined as follows (equivalent definition):
(P1) :
e0(p)→ min,
dx = f (x, u, t)dt + g(x, t)dµ, d∆k= 0, ∆k≥ 0, k = 0, 1,
By applying this maximum principle to (P1) and decoding its conditions in terms
of the data of problem (1.1)–(1.2), we conclude the existence of a function ψR ∈ Vn(T∗) such that dψR=−Hx(˜x, ψR, t)dt− Qx(˜x, ψR, t)d˜µ, t∈ T∗, ⎧ ⎪ ⎨ ⎪ ⎩ ψ0R= ξx(β0∗, t∗0,−∆∗0)∂e0 ∂x0 (p∗), ψ1R=−ξx(β1∗, t∗1, ∆∗1)∂e0 ∂x1 (p∗), (6.10) ξ∆(βk∗, t∗k, (−1)k+1∆∗k), ∂e0 ∂xk (p∗) = 0, k = 0, 1, (6.11) max u∈U(t)H(˜x(t), u, ψ R(t), t) = H(˜x(t), u∗(t), ψR(t), t), a.e., Q(˜x(t), ψR(t), t)≤ 0, t ∈ T∗, Q(˜x(t), ψR(t), t) = 0 ∀t ∈ supp(˜µ), ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ess lim sup
t→t∗k max u∈U(t)H(β ∗ k, u, ψ R k, t) +(−1)k ∂e0 ∂tk (p∗) + ξt(βk∗, t∗k, (−1) k+1∆∗ k), ∂e0 ∂xk (p∗) ≥ 0,
ess lim inf
t→t∗k umax∈U(t)H(β ∗ k, u, ψ R k, t) +(−1)k ∂e0 ∂tk (p∗) + ξt(βk∗, t∗k, (−1) k+1∆∗ k), ∂e0 ∂xk (p∗) ≤ 0, k = 0, 1.
Here, ˜x is the optimal trajectory for problem (P1), and the matrix function ξx and
the vector functions ξ∆, ξt are the derivatives of ξ with respect to, respectively, x,
∆, and t.
Clearly, ˜x(t) = x∗(t) ∀t ∈ (t∗0, t1∗), and ˜x(t∗k) = β∗k, k = 0, 1. Let us take ψ(t) =
ψR(t) ∀t ∈ (t∗0, t∗1), ψ(t∗k) = (−1)k ∂e0
∂xk(p
∗), k = 0, 1, and show that ψ satisfies (6.1)–
(6.5). Clearly, we need to verify them only at the two endpoints t∗k, k = 0, 1.
Consider the case k = 1 (the case k = 0 is similar). Let (α∗1, σ1) be a solution to
(6.7). It is straightforward to verify that function (ξx, ξ∆, ξt) is given by
(ξx, ξ∆, ξt)(β1∗, t∗1, ∆∗1) = ( ¯ξx, ¯ξ∆, ¯ξt)(1),
where ( ¯ξx, ¯ξ∆, ¯ξt)(0) = (E, 0, 0) (E is the n-dimensional identity matrix) and, for s∈ [0, 1], ⎧ ⎪ ⎨ ⎪ ⎩ ˙¯ ξx(s) = gx(α∗1(s), t∗1) ¯ξx(s)∆∗1, ˙¯ ξ∆(s) = gx(α∗1(s), t∗1) ¯ξ∆(s)∆∗1+ g(α∗1(s), t∗1), ˙¯ ξt(s) = gx(α∗1(s), t∗1) ¯ξt(s)∆∗1+ gt(α∗1(s), t∗1)∆∗1.
From (6.10) we get σ1(1) = ψ1. Thus (6.1)–(6.3) are proved.
To prove (6.4), we will show that σ1, g(α∗1, t∗1) ≡ 0. Indeed, by direct com-putation and by using the fact that dsdσ1, g(α1, t∗1) = 0 ∀s ∈ [0, 1], we have that
d
dsσ1, ¯ξ∆ = σ1, g(α∗1, t∗1) = const ∀s ∈ [0, 1]. On the other hand, from (6.11) and
the definition of ξ∆, we conclude thatσ1(0), ¯ξ∆(0) = σ1(1), ¯ξ∆(1) = 0.
Hence,σ1, ¯ξ∆ ≡ 0 and, as consequence, σ1, g(α∗1, t∗1) ≡ 0 on [0, 1], i.e., condi-tion (6.4) is proved.
To obtain inequalities in (6.5), note that these follow from the fact that
d
dsσ1, ¯ξt = σ1, gt(α ∗
1, t∗1) ∆∗1.
The condition (6.6) can be easily deduced as its analogue in (6.5). The proof is complete.
Theorem 6.2. Let (p∗, u∗, µ∗) be an optimal process for the problem specified by (1.1)–(1.3) and (1.5). Then, there exist a number λ0 ≥ 0, vectors λj ∈ kj, j = 1, 2, with|λ| = 1, and a function ψ ∈ Vn(T∗) satisfying
dψ =−Hx(t)dt− Qx(t)dµ∗, t∈ T∗, ψ0= ∂l ∂x0(p ∗, λ), ψ 1=− ∂l ∂x1(p ∗, λ), λ1≥ 0, λ1, e1(p∗) = 0; max u∈U(t) H(u, t) = H(t) a.e., Q(t)≤ 0 ∀t, Q(t) = 0 ∀t ∈ supp(µ∗), ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ess lim sup
t→t∗k max u∈U(t) H(β∗k, u, γk, t) +(−1)k+1 ∆∗k 1 0 gt(α∗k, t∗k), σk ds − ∂l ∂tk (p∗, λ) ≥ 0,
ess lim inf
t→t∗kumax∈U(t)H(β ∗ k, u, γk, t) +(−1)k+1 ∆∗k 1 0 gt(α∗k, t∗k), σk ds − ∂l ∂tk (p∗, λ) ≤ 0, k = 0, 1. (6.12)
Furthermore, inequalities (6.12) may be replaced by
ess lim inf
t→t∗k umax∈U(t)H(x ∗ k, u, ψk, t) + (−1)k ∂l ∂tk (p∗)≤ 0, k = 0, 1. (6.13)
However, we cannot guarantee that both conditions (6.12) and (6.13) hold at the same time, i.e., we have two different versions of the maximum principle.
Proof. The proof is based on the penalty function method [1]. Assume, first, that
∆∗k = 0, k = 0, 1. Take any natural i and define the function e0,i as follows: e0,i(p) = e0(p) +|p − p∗|2+ i(|e+1(p)|
2+|e2(p)|2).
Here, for a given vector a = (a1, . . . , ak), a+ stands for the vector with components
(aj)+= max{0, aj}, j = 1, . . . , k.
Let F (t) = F (t; µ∗) be the distribution function of the optimal measure. Take
c = gc supt∈T∗|x∗(t)| + |µ∗| + |p∗| + 1 and a sufficiently small ε > 0. Consider, for each i∈ N, the following penalty problem:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Ji(p, u, µ) = e0,i(p) +|y1− F (t∗1)| 2 +ε t1 t0 [|u − u∗(t)|2+|y − F (t)|2]dt→ min, dx = f (x, u, t)dt + g(x, t)dµ, t∈ [t0, t1], dy = dµ, y0= 0, t∈ [t0, t1], max{|p|, gc sup |x|, y1} ≤ c,
u(t)∈ U(t) a.e., µ ≥ 0.
(6.14)
For every i-problem, there exists a solution (pi, ui, µi).2 By using compactness
and by extracting a subsequence, we obtain a process (p, u, µ) such that pi → p, ui
w
→ u weakly in Lm 2(Tc), µi
w
→ µ weakly* in C∗(Tc).3 It is a straightforward task
to establish that p = p∗, u = u∗, and µ = µ∗. Moreover, from the penalty method we conclude that indeed ui → u∗ in Lm2(Tc). Then, by extracting a subsequence if
necessary, we conclude that ui(t)→ u∗(t) a.e.
All inequality constraints of problem (6.14) become strict for i sufficiently large. For this reason we can apply Theorem 6.1 to the i-problem. By writing down the necessary conditions of Theorem 6.1 and bearing in mind that i→ ∞, we conclude Theorem 6.2 when ∆∗k = 0, k = 0, 1. If ∆∗k > 0, then, by using reduction R1 (the
same way as in Theorem 6.1), we obtain the time-transversality conditions in the conventional case. To prove (6.13) we do not need to use reduction R1.
The proof is complete.
7. The problem with state constraints. In this section, we prove a weakened
maximum principle for problems with state and endpoint constraints.
Theorem 7.1. Consider the optimal control problem (1.1)–(1.5) with compatible
state and endpoint constraints and let triple (p∗, u∗, µ∗) be its solution.
Then, there exist a number λ0 ≥ 0, vectors λ1 ∈ k1, λ1 ≥ 0, λ2 ∈ k2, a
vector function ψ ∈ Vn(T∗), a vector measure η = (η1, . . . , ηk3), ηj ∈ C∗
+(T∗) such that Ds(µ∗)∩ Ds(ηj) = ∅ ∀j, and, for every atom r ∈ Ds(µ∗), there exist its own
vector function σr ∈ Vn([0, 1]) and its own vector measure ηr = (ηr1, . . . , ηkr3), ηrj ∈ C+∗([0, 1]), j = 1, . . . , k3, such that ψ(t) = ψ0− t t∗0 Hx(s)ds− [t∗0,t] Qx(s)dµ∗c + [t∗0,t] ϕx(x∗, s)dη + Σ(ψ, t), t∈ (t∗0, t∗1], (7.1) Σ(ψ, t) = r∈Ds(µ∗), r≤t [σr(1)− ψ(r−)], ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ˙ α∗r(s) = g(α∗r(s), r)∆∗r, s∈ [0, 1], dσr(s) =−gx(α∗r(s), r)σr(s)∆∗rds + ϕx(α∗r(s), r)dηr, s∈ [0, 1], α∗r(0) = x∗(r−), σr(0) = ψ(r−), ∆∗r = µ∗({r}), ψ0= ∂l ∂x0(p ∗, λ), ψ1=− ∂l ∂x1(p ∗, λ), (7.2) g(α∗ r(s), r), σr(s) = 0 ∀s ∈ [0, 1], ∀r ∈ Ds(µ∗) (7.3) supp(ηrj)⊆ {s ∈ [0, 1] : ϕj(αr∗(s), r) = Wj(α∗r(s), r) = 0} ∀j, (7.4) λ1, e1(p∗) = 0, ϕj(x∗(t), t) = 0 ηj-a.e. ∀j (7.5) max
u∈U(t)H(u, t) = H(t) a.e.,
(7.6)
2This is due to the compactness and Lemma 5.4. In fact, since x(t) are bounded uniformly (due
to gc sup|x| ≤ const and by using |µ| ≤ const) and Var |b
a[xi(t)] ≤ sups∈[a,b]|fi(s)| × |µi|([a, b])
∀a, b ∈ T (from the uniform bound of (ui, µi)), we can use Helly’s theorem and extract appropriate
subsequences.
3Here, T
Q(t)≤ 0 ∀t, Q(t) = 0 µ∗-a.e.,
(7.7)
ess lim inf
t→t∗k umax∈U(t) H(x∗k, u, ψk, t) + (−1)k ∂l ∂tk (p∗, λ)≤ 0, k = 0, 1, (7.8) |λ| + |η| + r∈Ds(µ∗) |ηr| = 1. (7.9)
Furthermore, when Assumption C is in force, then in place of (7.8) we can obtain time
transversality conditions, which are written as follows for the case where ∆∗k = 0:4
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ess lim sup
t→t∗k max u∈U(t)H(x ∗ k, u, ψk, t) + (−1)k ∂l ∂tk (p∗, λ)≥ 0,
ess lim inf
t→t∗k umax∈U(t) H(x∗k, u, ψk, t) + (−1)k ∂l ∂tk (p∗, λ)≤ 0, k = 0, 1. (7.10)
Besides, as in Theorem 6.2, we cannot guarantee that (7.8) and (7.10) are satisfied at the same time, so again we have two readings of the theorem.
Proof. The proof is based on a penalty function method [1].
Let us start by proving this result for the most difficult case, i.e., to consider the second variant of the theorem with time transversality conditions (7.10).
Bearing in mind Assumption C and Definition 2.3, let us take a sufficiently small
δ > 0. Since µ∗ is continuous at t∗k, k = 0, 1, there are points t0,δ> t∗0, t1,δ< t∗1, and
a small number ε > 0 such that (µ∗+L)([t∗0, t0,δ+ ε]∪ [t1,δ− ε, t∗1]) < δ. Define the function ϕ+δ as follows: ϕ+δ(x, t) = ϕ+(x, t), t 0,δ< t < t1,δ, 0 otherwise.
The components of ϕ+δ are nonnegative and lower semicontinuous.
Let F (t) = F (t; µ∗) and c = gc sup|x∗| + |p∗| + |µ∗| + 1. Take any i, A ∈ N and construct the following auxiliary penalty problem:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Ji,A(p, u, µ) = e0(p) +|p − p∗|2+|y1− F (t∗1)|2 +A t1 t0 |ϕ+ (x, t)|2dt + A [t0,t1] |ϕ+ δ(x, t)| 2 dµ + i−1 t1 t0 [|u − u∗(t)|2+|y − F (t)|2]dt→ min, dx = f (x, u, t)dt + g(x, t)dµ, dy = dµ, t∈ [t0, t1], y0= 0, e1(p)≤ 0, e2(p) = 0, |p − p∗| ≤ δ, µ([t0, t 0,δ+ ε)∪ (t1,δ− ε, t1])≤ δ,
max{gc sup |x|, y1} ≤ c, u(t)∈ U(t) a.e., µ ≥ 0.
(7.11)
The functional Ji,A is weakly lower semicontinuous in (u, µ) and, thus, for every i, A ∈ N, the problem (7.11) has a solution. Denote it by (pi,A, ui,A, µi,A) and let
(xi,A, yi,A) be the corresponding optimal trajectory. By using compactness, extract a
subsequence as i is fixed and obtain a process (p, u, µ) satisfying pi,A→ p, ui,A w → u weakly in Lm 2 (Tc), µi,A w → µ weakly* in C∗(Tc)5as A→ ∞. Furthermore, x i,A(t)→ x(t)∀t ∈ Cont(µ). We proceed by showing that p = p∗, u = u∗, µ = µ∗ (for every fixed i).
4Case ∆∗
k> 0 may be researched with the help of reduction R1. 5Again, T
Let us first show that (p, u, µ) is an admissible process. In fact, from (7.11), we have Ti,A |ϕ+(x i,A, t)|2dt + Ti,A |ϕ+ δ(xi,A, t)|2dµi,A≤ const A ,
and, by passing to the limit (with the help of Lemma 5.4) as A→ ∞, we obtain t1 t0 |ϕ+(x, t)|2dt + [t0,t1] |ϕ+ δ(x, t)| 2dµ = 0.
From here, and by using Assumption C and the compatibility condition, we deduce that gc sup ϕj(x)≤ 0, j = 1, . . . , k3. Thus, (p, u, µ) is an admissible process. Hence,
e0(p)≥ e0(p∗). Furthermore, since
Ji,A(pi,A, ui,A, µi,A)≤ Ji,A(p∗, u∗, µ∗) = e0(p∗),
we have that
e0(pi,A) +|pi,A− p∗|2+|y1,i,A− F (t∗1)| 2+ A Ti,A |ϕ+(x i,A, t)|2dt + A Ti,A |ϕ+ δ(xi,A, t)|2dµi,A+ 1 i Ti,A (|ui,A− u∗(t)|2+|yi,A− F (t)|2)dt≤ e0(p∗).
By passing to the limit in this inequality as A → ∞, and by using the weak lower semicontinuity of the left part in u, we obtain
e0(p) +|p − p∗|2+|y1− F (t∗1)|2+ 1 i t1 t0 (|u − u∗(t)|2+|y − F (t)|2)dt + lim sup A→∞ A Ti,A |ϕ+(x i,A, t)|2dt + A Ti,A |ϕ+ δ(xi,A, t)| 2dµ i,A ≤ e0(p∗). Since e0(p)≥ e0(p∗), we conclude immediately that p = p∗, u = u∗, µ = µ∗, and
lim A→∞A Ti,A |ϕ+(x i,A, t)|2dt + A Ti,A |ϕ+
δ(xi,A, t)|2dµi,A= 0 ∀i.
So, for every i, we can take a number Ai satisfying |pi,Ai− p ∗|2+u i,Ai− u ∗2 L2+yi,Ai− F (t) 2 L2 + Ai Ti,Ai |ϕ+(x i,Ai, t)| 2dt + A i Ti,Ai |ϕ+ δ(xi,Ai, t)| 2dµ i,Ai ≤ 1 i
and define a new diagonal sequence denoted by (pi, ui, µi), i.e., pi= pi,Ai, ui= ui,Ai, µi= µi,Ai. (7.12)
By extracting a subsequence, we obtain pi → p∗, ui → u∗ a.e., µi w
→ µ∗ as i→ ∞.
For this sequence, we have xi(t) = xi,Ai(t) → x∗(t) ∀t ∈ Cont(µ∗). By replacing
A in problem (7.11) by this specially chosen Ai, a new problem, called i-problem, is
All the additional inequality constraints of i-problem become strict inequalities for i sufficiently large. Therefore, Theorem 6.2 can be applied to the i-problem. It states that for every such i, there are functions ψi ∈ Vn(Ti), ζi ∈ C(Ti), a number λ0,i≥ 0, and vectors λ1,i and λ2,isuch that
dψi=−Hxi(t)dt− Q i x(t)dµi+ ϕx(xi, t)dηi, t∈ Ti, (7.13) ψ0,i= ∂l ∂x0
(pi, λi) + 2λ0,i(x0,i− x∗0), ψ1,i=− ∂l ∂x1 (pi, λi)− 2λ0,i(x1,i− x∗1), (7.14) dζi= 2λ0,i
i [yi− F (t)]dt, ζi(t1,i) =−2λ0,i[y1,i− F (t ∗ 1)], λ1,i≥ 0, e1(pi), λ1,i = 0,
max
u∈U(t)[Hi(u, t)− λ0,ii
−1|u − u∗(t)|2] = H i(t)− λ0,ii−1|ui(t)− u∗(t)|2 a.e., (7.15) Qi(t) + ζi(t)− λ0,iAi|ϕ+δ(xi(t), t)|2≤ 0 ∀t, (7.16) Qi(t) + ζi(t)− λ0,iAi|ϕδ+(xi(t), t)|2= 0 ∀t ∈ supp(µi), (7.17) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ess lim sup
t→tk,i max
u∈U(t)
[H(βk,i, u, γk,i, t)− λ0,ii−1|u − u∗(t)|2] −λ0,iAi|ϕ+(βk,i, tk,i)|2+ (−1)k
∂l ∂tk (pi, λi) + 2λ0,i(tk,i− t∗k) ≥ 1(i),
ess lim inf
t→tk,i max
u∈U(t)[H(βk,i, u, γk,i, t)− λ0,ii
−1|u − u∗(t)|2] −λ0,iAi|ϕ+(βk,i, tk,i)|2+ (−1)k
∂l ∂tk (pi, λi) + 2λ0,i(tk,i− t∗k) ≤ 1(i), k = 0, 1, (7.18) |λi| + gc sup t∈Ti |ψi(t)| + |ηi| = 1. (7.19)
Here, Ti = [t0,i, t1,i], λi = (λ0,i, λ1,i, λ2,i), β0,i = xi(t+0,i), β1,i = xi(t−1,i), γ0,i = ψi(t+0,i), γ1,i= ψi(t−1,i), and ηi= (η1i, . . . , η
k3
i ) is a vector measure whose components
are defined, j = 1, . . . , k3, by the distribution functions
F (t; ηij) = 2λ0,iAi [t0,i,t] [ϕj(xi, t)]+ds + 2λ0,iAi [t0,i,t] [ϕjδ(xi, t)]+dµi, t > t0,i.
The expression 1(i) denotes a sequence of numbers converging to zero as i→ ∞, and the index i in H and Q (and its partial derivatives) indicates that they are evaluated at xi(t), ui(t), and ψi(t), whenever the corresponding arguments x, u, and ψ are
missing.
Note that because the function ϕ+δ is discontinuous, the i-problem is a nonstan-dard one. However, Theorem 6.2 applies in the same way for this problem. Further-more, conditions (7.13)–(7.19) hold independently of the values µi({tk,δ}), k = 0, 1.
This is so due to the nonnegativity and lower semicontinuity of the function |ϕ+δ|2
and also to the following fact: if|ϕ+|2 is zero at some point (x, t), then its derivative
is zero at this point.
From (7.19), we get that ζi → 0 uniformly. Again from (7.19), and by using
compactness, we conclude that, after subsequence extraction, λi → λ, and η j i w → ˜ ηj, j = 1, . . . , k
3, as i → ∞. Let, for every Borel set B ⊆ 1, η(B) = ˜η(B)−
˜
η(B∩ Ds(µ∗)), where ˜η = (˜η1, . . . , ˜ηk3). Thus, η = (η1, . . . , ηk3) is a vector measure
Also from (7.19), it follows that the variation of ψi is bounded uniformly in i.
By using the second Helly theorem and by extracting a subsequence, we obtain a function ψH such that ψi(t) → ψH(t) ∀t ∈ T∗ as i → ∞. Let us find ψ ∈ Vn(T∗)
such that ψ(t) = ψH(t)∀t ∈ Cont(µ∗)∩ Cont(η).6 Such function ψ(t) exists because
of the estimate Var|b
a[ψH]≤ c(µ∗+ Var ˜η +L)([a, b]) ∀a ≤ b, which follows from the
inequality Var ν[ψi]≤ c(µi+ Var ηi+L) as i → ∞.
Let us prove that ψ(t) satisfies the conditions of our theorem. For this purpose, let us construct a countable family of absolutely continuous measures {ˆµi}, i ∈ N,
which approximately satisfies all the conditions (except, possibly, the ones concerning time transversality (7.18)) of the maximum principle for the i-problem, as follows.
Let{tk}, k ∈ N, be a countable set of points everywhere dense in T∗ such that tk ∈ X = Cont(µ∗)∩ Cont(η) ∩ [
∞
i=1Cont(µi)],
7 and let {φ
k} be a countable set
of functions everywhere dense in C(Tc). For every i sufficiently large, there exists a
sequence of absolutely continuous measures µi,τ having densities mi,τ > 0 a.e., such
that µi,τ w → µi on Ti, µi,τ w → µi on [t0,i, t0,δ], µi,τ w → µi on [t1,δ, t1,i] as τ → ∞ for all τ .
Let the pair (xi,τ, ψi,τ) satisfy the following differential system:8
⎧ ⎨ ⎩
dxi,τ = f (xi,τ, ui, t)dt + g(xi,τ, t)dµi,τ,
dψi,τ =−Hx(xi,τ, ui, ψi,τ, t)dt− Qx(xi,τ, ψi,τ, t)dµi,τ+ ϕx(xi,τ, t)dηi,τ, xi,τ(t0,i) = xi(t0,i), ψi,τ(t0,i) = ψi(t0,i),
where the measure ηji,τ is given by its distribution function
F (t; ηi,τj ) = 2λ0,iAi [t0,i,t] [ϕj(xi,τ, t)]+ds + 2λ0,iAi [t0,i,t] [ϕjδ(xi,τ, t)]+dµi,τ, j = 1, . . . , k3.
By using Lemma 5.4, choose a number τi such that
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ k3 j=1 i k=1 T i φk(t)d(µi,τi− µi) +T i φk(t)d(ηi,τj i− η j i) ≤ 1 i, i k=1 |ψi,τi(tk)− ψi(tk)| + gc sup t∈Ti |ψi(t)| − max t∈Ti |ψi,τi(t)| ≤ 1i, T i ˜
Qi(xi,τi, ψi,τi, t)dµi,τi− Ti ˜ Qi(xi, ψi, t)dµi + k3 j=1 Ti ϕj(xi,τi, t)dη j i,τi− Ti ϕj(xi, t)dηij ≤ 1 i, |ψi,τi(t1,i)− ψi(t1,i)| + maxt
∈Ti | ˜Qi(xi,τi, ψi,τi, t)− ˜Qi(xi, ψi, t)| ≤ 1 i, (7.20) where ˜Qi(x, ψ, t) = Q(x, ψ, t)− λ0,iAi|ϕ+δ(x, t)| 2.
6From now on, Cont(η) =k3 j=1Cont(η
j).
7Such an everywhere dense set exists, because, being a countable union of countable sets, T∗\ X
is countable.
Put ˆψi= ψi,τi, ˆxi= xi,τi, ˆµi = µi,τi, ˆηi= ηi,τi. By extracting a subsequence, we have that ˆψi(t)→ ψA(t)∀t ∈ T∗.
Now, we show that ψA(t) = ψ(t) ∀t ∈ Cont(µ∗) ∩ Cont(η). Indeed, since
Var ν[ψi]≤ c(µi+Var ηi+L), ψ(t) is continuous on the set Cont(µ∗)∩Cont(η)\{t∗0, t∗1}.
The same can be said about ψA. Then, it follows from (7.20) that ψA(t) = ψ(t) ∀t ∈ Cont(µ∗)∩Cont(η). For such a sequence, ˆxi(t)→ x∗(t),∀t ∈ Cont(µ∗), ˆµi w
→ µ∗,
ˆ
ηij→ ˜ηw j, j = 1, . . . , k
3, as i→ ∞.
Let us prove (7.1). We start by showing that there are sequences of absolutely continuous measures{¯µi}, {¯ηji}, j = 1, . . . , k3, and also a sequence of natural numbers ki≥ i such that (1) ¯µi w → µ∗ d, and, ∀j, ¯η j i w → ˜ηj− ηj as i→ ∞, (2) ˆµki ≥ ¯µi, ˆη j ki≥ ¯η j i ∀i, j. If Ds(µ∗) =∅, then we put ¯µi= ¯η j i = 0, ki= i,∀i, j.
Let Ds(µ∗) = ∅. Consider the chain of sets Di such that D0 =∅, Di−1 ⊆ Di ⊆
Ds(µ∗) and r∈Ds(µ∗)\Diµ
∗({r}) ≤ 1
i, i ∈ N. Define sets Sr,i = [r− ρi, r + ρi], r∈ Di, as a system of pairwise disjoint closed neighborhoods of points r such that
(i) ρi > 0, ρi→ 0 as i → ∞, (ii) |µ∗(Si)− r∈Diµ ∗({r})| +k3 j=1|˜η j(S i)− r∈Diη˜ j({r})| ≤1 i, where Si= r∈DiSr,i,
(iii) r± ρi∈ Cont(µ∗)∩ Cont(η).
The existence of such sets Si follows from the regularity of Borel measures µ∗, ˜ηj, j = 1, . . . , k3. Due to the weak star convergence of the considered sequences of
measures, it is possible to take a natural number ki≥ i such that
r∈Di ˆµki(Sr,i)− µ ∗(S r,i) +k3 j=1 ˆηj ki(Sr,i)− ˜η j(S r,i) ≤1 i. Define ¯µi and ¯ηi={¯η1i, . . . , ¯η k3 i }, respectively, by ¯µi(B) := ˆµki(B∩ Si), and ¯ηi(B) := ˆ
ηki(B∩ Si), for any Borel set B ⊆
1. Clearly, ¯µ i w → µ∗ d, ¯η j i w → ˜ηj − ηj as i → ∞ (and, hence, ˆµki− ¯µi w → µ∗ c, ˆη j ki− ¯η j i w → ηj) for j = 1, . . . , k
3. It is also clear that
ˆ µki− ¯µi∈ C+∗(Tc), ˆη j ki− ¯η j i ∈ C+∗(Tc)∀i, j.
From now on, consider subsequences extracted from{ˆµi}, {ˆηi}, {ˆxi}, { ˆψi}, {ui},
and{Ti} (denote them again by the old index i) according to the constructed sequence {ki} and let us rewrite (1.2), (7.13) as follows:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˆ xi(t) = x0,i+ t t0,i f (ˆxi, ui, s)ds + [t0,i,t] g(ˆxi, s)d(ˆµi− ¯µi) + [t0,i,t] g(ˆxi, s)d¯µi, ˆ ψi(t) = ψ0,i− t t0,i ˆ Hxi(s)ds− [t0,i,t] ˆ Qix(s)d(ˆµi− ¯µi)− [t0,i,t] ˆ Qix(s)d¯µi + [t0,i,t] ϕx(ˆxi, s)d(ˆηi− ¯ηi) + [t0,i,t] ϕx(ˆxi, s)d¯ηi, t∈ Ti.
From now on, the index i and the sign hat in H and Q (and also its partial derivatives) stand for their values when ˆxi(t), ui(t), and ˆψi(t) replace the omitted arguments x,
u, and ψ respectively. Put,∀t ∈ Ti, ˆ xci(t) = x0,i+ t t0,i f (ˆxi, ui, s)ds + [t0,i,t] g(ˆxi, s)d(ˆµi− ¯µi), ˆ xd i(t) = [t0,i,t] g(ˆxi, s)d¯µi, ˆ ψηi(t) = ψ0,i− t t0,i ˆ Hxi(s)ds− [t0,i,t] ˆ Qix(s)d(ˆµi− ¯µi) + [t0,i,t] ϕx(ˆxi, s)d(ˆηi− ¯ηi), ˆ ψd i(t) =− [t0,i,t] ˆ Qix(s)d¯µi+ [t0,i,t] ϕx(ˆxi, s)d¯ηi. Thus, ˆxi(t) = ˆxci(t) + ˆxdi(t), ˆψi(t) = ˆψ η i(t) + ˆψ d
i(t) ∀t ∈ Ti. Bearing in mind that
ˆ
xi(t)→ x∗(t)∀t ∈ Cont(µ∗), by Lebesgue theorem and Lemma 5.1, we deduce that
ˆ xci(t)→ x∗c(t) = x∗0+ t t∗0 f (x∗, u∗, s)ds + [t∗0,t] g(x∗, s)dµ∗c ∀t ∈ T∗, i→ ∞.
In a similar way, from Lemma 5.1, we conclude that∀t ∈ Cont(η), ˆ ψηi(t)→ ψη(t) = ψ0− t t∗0 Hx(s)ds− [t∗0,t] Qx(s)dµ∗c+ [t∗0,t] ϕx(x∗, s)dη
as i→ ∞. Let ψd= ψ− ψη. Then, ˆψid(t)→ ψd(t)∀t ∈ Cont(µ∗)∩ Cont(η).
Let us show ψd = Σ(ψ, t). In fact, fix t∈ Cont(µ∗)∩ Cont(η) and ε > 0. Take a
number N = N (ε) such thatr∈Ds(µ∗)\DN[µ
∗({r}) + Var ˜η({r})] ≤ ε. In this case,
then lim sup i→∞ r∈D(N,t) Sr,i − ˆQix(s)d¯µi+ Sr,i ϕx(ˆxi, s)d¯ηi − ˆψid(t) ≤ const ε, (7.21)
where, from now on, D(N, t) ={r ∈ DN : r≤ t}.
Let r−i = r− ρi and, for r∈ D(N, t), consider, on segment Sr,i, the system
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ˆ xdi(s) = ˆxdi(r−i ) + [ri−,s] g(ˆxi(τ ), τ )d¯µi, ˆ ψid(s) = ˆψid(ri−)− [r−i,s] gx(ˆxi(τ ), τ ) ˆψi(τ )d¯µi+ [ri−,s] ϕx(ˆxi(τ ), τ )d¯ηi. (7.22)
For each i sufficiently large, we define the function9
πr,i(τ ) =
F (τ ; ¯µi)− F (r−i ; ¯µi)
¯
µi(Sr,i)
, r∈ D(N, t), τ ∈ Sr,i.
The function πr,i: Sr,i→ [0, 1] is absolutely continuous and strictly increasing, dπdτr,i = ¯
mi(τ )
¯
µi(Sr,i) > 0 ( ¯mi(τ ) is the density of measure ¯µi). Hence, there exists an inverse mapping, θr,i : [0, 1] → Sr,i, θr,i = (πr,i)−1, which is also absolutely continuous and
9Let us take numbers i such that t /∈ S
