Error Estimates for the Euler Discretization of an Optimal Control Problem with First-Order State Constraints
Abstract
We propose some error estimates for the discrete solution of an optimal control problem with first-order state constraints, where the trajectories are approximated with a classical Euler scheme. We obtain order 1 approximation results in the $L^\infty$ norm (as opposed to the order 2/3 results obtained in the literature). We assume either a strong second-order optimality condition or a weaker formulation in the case where the state constraint is scalar and satisfies some hypotheses for junction points, and where the time step is constant. Our technique is based on some homotopy path of discrete optimal control problems that we study using perturbation analysis of nonlinear programming problems.
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References
1.
J. F. Bonnans, Lipschitz solutions of optimal control problems with state constraints of arbitrary order, Ann. Acad. Rom. Sci. Ser. Math. Appl., 2 (2010), pp. 78--98.
2.
J. F. Bonnans, J. C. Gilbert, C. Lemarechal, and C. A. Sagastizábal, Numerical Optimization: Theoretical and Practical Aspects, Springer, Berlin, 2006.
3.
J. F. Bonnans and A. Hermant, Well-posedness of the shooting algorithm for state constrained optimal control problems with a single constraint and control, SIAM J. Control Optim., 46 (2007), pp. 1398--1430.
4.
J. F. Bonnans and A. Hermant, Stability and sensitivity analysis for optimal control problems with a first-order state constraint and application to continuation methods, ESAIM Control Optim. Calc. Var., 14 (2008), pp. 825--863.
5.
J. F. Bonnans and A. Hermant, Revisiting the analysis of optimal control problems with several state constraints, Control Cybernet., 38 (2009), pp. 1021--1052.
6.
J. F. Bonnans and A. Hermant, Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), pp. 561--598.
7.
J. F. Bonnans and A. Hermant, No-gap second-order optimality conditions for optimal control problems with a single state constraint and control, Math. Program, 117 (2009), pp. 21--50.
8.
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer, New York, 2000.
9.
F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, RAIRO Oper. Res., 8.2 (1974), pp. 129--151.
10.
A. E. Bryson, W. F. Denham, and S. E. Dreyfus, Optimal programming problems with inequality constraints, AIAA J., 1 (1963), pp. 2544--2550.
11.
B. M. Budak, E. M. Berkovich, and E. N. Solov'eva, Difference approximations in optimal control problems, SIAM J. Control, 7 (1969), pp. 18--31.
12.
J. Cullum, Finite-dimensional approximations of state-constrained continuous optimal control problems, SIAM J. Control, 10 (1972), pp. 649--670.
13.
A. L. Dontchev, Discrete approximations in optimal control, in Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control, IMA Vols. Math. Appl. 78, Springer, New York, 1996, pp. 59--80.
14.
A. L. Dontchev and W. W. Hager, Lipschitzian stability in nonlinear control and optimization, SIAM J. Control Optim., 31 (1993), pp. 569--603.
15.
A. L. Dontchev and W. W. Hager, Lipschitzian stability for state constrained nonlinear optimal control, SIAM J. Control Optim., 36 (1998), pp. 698--718.
16.
A. L. Dontchev and W. W. Hager, The Euler approximation in state constrained optimal control, Math. Comp., 70 (2001), pp. 173--203.
17.
W. W. Hager, The Ritz--Trefftz method for state and control constrained optimal control problems, SIAM J. Numer. Anal., 12 (1975), pp. 854--867.
18.
W. W. Hager, Lipschitz continuity for constrained processes, SIAM J. Control Optim., 17 (1979), pp. 321--338.
19.
W. W. Hager, Runge--Kutta methods in optimal control and the transformed adjoint system, Numer. Math., 87 (2000), pp. 247--282.
20.
W. W. Hager and G. D. Ianculescu, Dual approximations in optimal control, SIAM J. Control Optim., 22 (1984), pp. 423--465.
21.
A. Hermant, Stability analysis of optimal control problems with a second-order state constraint, SIAM J. Optim., 20 (2009), pp. 104--129.
22.
D. H. Jacobson, M. M. Lele, and J. L. Speyer, New necessary conditions of optimality for control problems with state-variable inequality constraints, J. Math. Anal. Appl., 35 (1971), pp. 255--284.
23.
K. Jittorntrum, Solution point differentiability without strict complementarity in nonlinear programming, in Sensitivity, Stability and Parametric Analysis, Mathematical Programming Studies 21, Springer, Berlin, 1984, pp. 127--138.
24.
K. Malanowski and H. Maurer, Sensitivity analysis for state constrained optimal control problems, Discrete Contin. Dyn. Syst., 4 (1998), pp. 241--272.
25.
H. Maurer, On the minimum principle for optimal control problems with state constraints, Technical report, Schriftenreihe des Rechenzentrum 41, Universität Münster, 1979.
26.
H. Maurer, First and second order sufficient optimality conditions in mathematical programming and optimal control, in Mathematical Programming at Obewogach, Mathematical Programming Studies 14, Sprniger, Berlin, 1981, pp. 163--177.
27.
H. Maurer and H. J. Pesch, Solution differentiability for nonlinear parametric control problems, SIAM J. Control Optim., 32 (1994), pp. 1542--1554.
28.
H. Maurer and J. Zowe, First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems, Math. Program., 16 (1979), pp. 98--110.
29.
A. A. Milyutin and N. P. Osmolovskii, Calculus of Variations and Optimal Control, American Mathematical Society, Providence, RI, 1998.
30.
B. Sh. Mordukhovich, On difference approximations of optimal control systems, J. Appl. Math. Mech., 42 (1978), pp. 452--461.
31.
L. W. Neustadt, A general theory of extremals, J. Comput. Syst. Sci. Int., 3 (1969), pp. 57--92.
32.
S. M. Robinson, First order conditions for general nonlinear optimization, SIAM J. Appl. Math., 30 (1976), pp. 597--607.
33.
S. M. Robinson, Strongly regular generalized equations, Math. Oper. Res., 5 (1980), pp. 43--62.
34.
A. Schwartz and E. Polak, Consistent approximations for optimal control problems based on Runge--Kutta integration, SIAM J. Control Optim., 34 (1996), pp. 1235--1269.
35.
R. Vinter, Optimal Control, Birkhäuser, Boston, MA, 2000.
36.
J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim., 5 (1979), pp. 49--62.
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© 2017, Society for Industrial and Applied Mathematics.
History
Submitted: 10 December 2014
Accepted: 23 November 2016
Published online: 2 March 2017
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Seventh Framework Programme http://dx.doi.org/10.13039/501100004963 : FP7-PEOPLE-2010-ITN- SADCO
Université Paris-Saclay http://dx.doi.org/10.13039/501100007241 : PGMO program of the Fondation Mathematique Jacques Hadamard
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