University of Padova, Padova, Italy CONSTRAINT SOLUTIONS (on input-output evaluations for unconstrained controls) joint work with (IMPA) 2 / 37New trends in optimal control. University of Padova, Padova, Italy CONSTRAINT SOLUTIONS (on input-output evaluations for unconstrained controls) joint work with (IMPA) 4 / 37New trends in optimal control. University of Padova, Padova, Italy CONSTRAINT SOLUTIONS (on input-output evaluations for unconstrained controls) joint work with (IMPA) 5 / 37New trends in optimal control.
University of Padova, Padova, Italy CONSTRAINT SOLUTIONS (on input-output evaluations for unconstrained controls) joint work with (IMPA) 7 / 37New trends in optimal control. University of Padova, Padova, Italy CONSTRAINT SOLUTIONS (on input-output evaluations for unconstrained controls) joint work with (IMPA) 8 / 37New trends in optimal control.
EXISTING NOTIONS OF SOLUTION FOR
University of Padova, Padova, Italy CONSTRAINT SOLUTIONS (on input-output evaluations for unconstrained controls) joint work with (IMPA) 9 / 37New trends in optimal control. University of Padova, Padova, Italy CONSTRAINT SOLUTIONS (on input-output evaluations for unconstrained controls) joint work with (IMPA) 10 / 37New trends in optimal control. University of Padova, Padova, Italy CONSTRAINT SOLUTIONS (on input-output evaluations for unconstrained controls) joint work with (IMPA) 12 / 37New trends in optimal control.
University of Padua, Padua, Italy CONSTRAINED SOLUTIONS (on I/O extensions for unconstrained controls) joint work with (IMPA) 13 / 37 New Trends in Optimal Control. University of Padua, Padua, Italy CONSTRAINED SOLUTIONS (on I/O extensions for unconstrained controls) joint work with (IMPA) 14 / 37 New Trends in Optimal Control. University of Padua, Padua, Italy CONSTRAINED SOLUTIONS (on I/O extensions for unconstrained controls) joint work with (IMPA) 16 / 37 New Trends in Optimal Control.
University of Padua, Padua, Italy LIMIT SOLUTIONS (on extensions of input-output for unbounded controls) joint work with (IMPA) 18 / 37 New trends in Optimal Control. University of Padua, Padua, Italy LIMIT SOLUTIONS (on extensions of input-output for unbounded controls) joint work with (IMPA) 20 / 37 New trends in Optimal Control. University of Padua, Padua, Italy LIMIT SOLUTIONS (on extensions of input-output for unbounded controls) joint work with (IMPA) 21 / 37 New trends in Optimal Control.
Some requirements must be met:. consistency with the Carath'eodoric notion of solution foru ∈AC ; x single value on each;. and possibly uniqueness) results previous definitions of solution for impulsive systems should be subordinated to this extended concept. University of Padova, Padova, Italy CONSTRAINT SOLUTIONS (on input-output evaluations for unconstrained controls) joint work with (IMPA) 21 / 37New trends in optimal control. University of Padova, Padova, Italy CONSTRAINT SOLUTIONS (on input-output evaluations for unconstrained controls) joint work with (IMPA) 22 / 37New trends in optimal control.
University of Padova, Padova, Italy CONSTRAINED SOLUTIONS (on determinations of input-output for unconstrained controls) joint work with (IMPA) 23 / 37New trends in optimal control. Continuous dependence: for each τ ∈[a,b] one has. moreover: one has constant dependence w.r. LIMIT” SOLUTIONS Proposed definition of Limit Solution. University of Padova, Padova, Italy CONSTRAINED SOLUTIONS (on determinations of input-output for unconstrained controls) joint work with (IMPA) 24 / 37 New trends in optimal control.
University of Padua, Padua, Italy LIMIT SOLUTIONS (on extensions of input-output for unbounded controls) joint work with (IMPA) 25 / 37 New trends in Optimal Control.
THE GENERIC, NON COMMUTATIVE, CASE
University of Padova, Padova, Italy CONSTRAINED SOLUTIONS (on determinations of input-output for unconstrained controls) joint work with (IMPA) 26 / 37New trends in optimal control. University of Padova, Padova, Italy CONSTRAINED SOLUTIONS (on determinations of input-output for unconstrained controls) joint work with (IMPA) 27 / 37New trends in optimal control. University of Padova, Padova, Italy CONSTRAINED SOLUTIONS (on determinations of input-output for unconstrained controls) joint work with (IMPA) 28 / 37New trends in optimal control.
University of Padova, Padova, Italy CONSTRAINT SOLUTIONS (on input-output evaluations for unconstrained controls) joint work with (IMPA) 29 / 37New trends in optimal control. University of Padova, Padova, Italy CONSTRAINT SOLUTIONS (on input-output evaluations for unconstrained controls) joint work with (IMPA) 30 / 37New trends in optimal control. University of Padova, Padova, Italy CONSTRAINT SOLUTIONS (on input-output evaluations for unconstrained controls) joint work with (IMPA) 31 / 37New trends in optimal control.
University of Padova, Padova, Italy CONSTRAINT SOLUTIONS (on input-output evaluations for unconstrained controls) joint work with (IMPA) 32 / 37New trends in optimal control. Note: In the previous example, the fact that xˆ is a smooth simple boundary solution does not mean that x is a Carath´eodoric solution. University of Padova, Padova, Italy CONSTRAINT SOLUTIONS (on input-output evaluations for unconstrained controls) joint work with (IMPA) 33 / 37New trends in optimal control.
Corresponding extension for minimal problems (Aronne's next talk) Classify other classes of solutions with unbounded variation. University of Padua, Padua, Italy LIMIT SOLUTIONS (on extensions of input-output for unbounded controls) joint work with (IMPA) 34 / 37 New trends in Optimal Control. University of Padua, Padua, Italy LIMIT SOLUTIONS (on extensions of input-output for unbounded controls) joint work with (IMPA) 36 / 37 New trends in Optimal Control.
University of Padua, Padua, ItalyLIMIT SOLUTIONS (on extensions of input-output for unbounded controls) joint work with (IMPA) 37 / 37New trends in Optimal Control. University of Padua, Padua, ItalyLIMIT SOLUTIONS (on extensions of input-output for unbounded controls) joint work with (IMPA) 37 / 37New trends in Optimal Control.
