Figure 3.3 – Case A.L1-minimum trajectory. The graph displays the trajectory (solid curve) as well as the action of the control (arrows). The initial orbit is strongly eccentric (the eccen- tricity equals 0.75) and strongly inclined (the inclination is 56degrees). The geostationary target orbit around the Earth is reached attf »147.28hours. The sparse structure of the con- trol is clearly observed, with burn arcs concentrated around perigees and apogees (see [31]).
The minimization leads to thrust only35%of the time.
0 50 100 150 200 250 300
−0.5 0 0.5
Figure 3.4 – Case A. Conjugate point test on the bang-bangL1-extremal extended tor0,2tfs.
The value of the determinant of Jacobi fields along the extremal is plotted against time. The first conjugate point occurs att1c «171.20hours°tf; optimality of the reference extremal on r0, tfsfollows. Jumps on the Jacobi fields are observed at each switching time, and conjugacy occurs at such a switching (sign change of the determinant).
3.4. NUMERICAL EXAMPLES FOR PROBLEMS WITH FIXED ENDPOINTS: THE TWO-BODY CASE
a convergence result as in [85], and to verify the second order conditions on the sequence of regularized extremals. As underlined in Sect. 3.1, conjugate times may occur at or between switching times. On the example treated, no conjugate point is detected on r0, tfs, ensuring a strong local optimum (cf. Theorem 3.1). The extremal is then extended up to2tf, and a conjugate point is detected about 1.1tf, at a switching point (sign change occurring at the jump).
Remark 3.3. AsH0is the lift of a vector field, the determinant of Jacobi fields is either iden- tically zero or non-vanishing along a cost arc (⇢“0). (Compare with the case of polyhedral control set; see also Corollary 3.9 in [63].) Moreover, coming from the two-body case, the driftf0 is the symplectic gradient of the energy function,
Epr,vq:“ 1
2|v|2´ 1
|r|2.
Accordingly, the x“ p r, vqpart of the Jacobi fields along an integral arc of›ÑH0 verifies 9
xptq “›ÑE1pxptqq xptq,
so xhas a constant determinant along such an arc since the associated flow is symplectic.
In particular, the disconjugacy condition (or Conditions3.1and3.2) implies that the optimal solution starts with a burn arc.
3.4.2 Case B
In case B, we compute the transfer problem with a different inclination and a different max- imum thrust (see Tab. 3.2). Using the same numerical method as in case A, the optimal
Table 3.2 – Case B. Summary of physical constants used for the numerical computation.
Gravitational constantµof the Earth: 398600.47Km3s´2 Mass of the spacecraft: 1500Kg Thrust: 10Newtons Initial perigee: 6643Km Final perigee: 42165Km
Initial apogee: 46500Km Final apogee: 42165Km
Initial inclination: 7deg Final inclination: 0deg Initial longitude: ⇡rad Final longitude: 56.659rad Minimum time: 110.41hours Fixed final time: 147.28hours
L1 cost achieved (normalized): 67.617
solution is computed and displayed in Fig. 3.5. The piecewise continuous function ptq is computed by the numerical procedure in Appendix C. The extremal is then extended up to
Figure 3.5 – Case B. L1 minimum trajectory. The graph displays the trajectory (blue line), as well as the action of the control (red arrows). The initial orbit is strongly eccentric (0.75) and slightly inclined (7degrees). The geostationary target orbit around the Earth is reached attf » 147.28hours. The sparse structure of the control is clearly observed, with burn arcs concentrated around perigees and apogees (see [31]). The minimization leads to thrust only 46% of the time. This percentage is higher than for case A (compared with Fig.3.3), which is qualitatively consistent with the fact that the ratio of the final time vs. the minimum time is diminished in Case B.
0 200 400
#10-8
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1
489 490 491
#10-14
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Figure 3.6 – Case B. Conjugate point test on a perturbed bang-bang L1-extremal extended to r0,3.5tfs. The value of the determinant of Jacobi fields along the extremal is plotted against time (detail on the right subgraph). The endpoint conditionsx0, xf given in Tab.3.1 are perturbed according to x – x` x, | x| » 1e´5, leading to conjugacy not at but between switching points—along a burn arc (⇢ “ 1). The first conjugate point occurs at t1c »489.23hours°tf, ensuring again optimality of the reference extremal onr0, tfs.
3.5tf, and a conjugate point is detected about 3.2tf at a switching point (see Fig. 3.7). A test on a perturbation of case B is provided in Fig. 3.6; by slightly changing the endpoint
3.4. NUMERICAL EXAMPLES FOR PROBLEMS WITH FIXED ENDPOINTS: THE TWO-BODY CASE
0 100 200 300 400 500
#10-8
-4 -2 0 2
475 476 477 478
#10-13
-1 -0.5 0 0.5 1 1.5
0 50 100 150 200 250 300 350 400 450 500
0 0.2 0.4 0.6 0.8 1
Figure 3.7 – Conjugate point test on the bang-bang L1-extremal extended tor0,3.5tfs. The value of the determinant of Jacobi fields along the extremal is plotted against time on the upper left subgraph. The first conjugate point occurs att1c » 475.93hours° tf; optimality of the reference extremal onr0, tfsfollows. On the upper right subgraph, a zoom is provided to show the jumps on the Jacobi fields (then on their determinant) around the first conjugate time; several jumps are observed, the first one leading to a sign change at the conjugate time.
Note that in accordance with Remark3.3, the determinant must be constant along the cost arcs (⇢ “0) provided the symplectic coordinatesx“ pr,vqare used; this is not the case here as the so called equinoctial elements [20] are used for the state—hence the slight change in the determinant. The bang-bang norm of the control, rescaled to belong tor0,1sand extended to 3.5tf, is portrayed on the lower graph. On the extended time span, there are already more than 70switchings though the thrust is just a medium one. For low thrusts, hundreds of switchings occur.
conditions, one observes that conjugacy occurs not at a switching anymore, but along a burn (maximum-thrust) arc.