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Optimality, stability, and convergence in nonlinear control. (English) Zbl 0821.49022

Summary: Sufficient optimality conditions for infinite-dimensional optimization problems are derived in a setting that is applicable to optimal control with endpoint constraints and with equality and inequality constraints on the controls. These conditions involve controllability of the system dynamics, independence of the gradients of active control constraints, and a relatively weak coercivity assumption for the integral cost functional. Under these hypotheses, we show that the solution to an optimal control problem is Lipschitz stable relative to problem perturbations. As an application of this stability result, we establish convergence results for the sequential quadratic programming algorithm and for penalty and multiplier approximations applied to optimal control problems.

MSC:

49K40 Sensitivity, stability, well-posedness
49M30 Other numerical methods in calculus of variations (MSC2010)
90C20 Quadratic programming
Full Text: DOI

References:

[1] Alt W (1989) Stability of solutions for a class of nonlinear cone constrained optimization problems, Part 1: Basic theory. Numer Funct Anal Optim 10:1053-1064 · Zbl 0679.49026 · doi:10.1080/01630568908816346
[2] Alt W (1990) Stability of solutions to control constrained nonlinear optimal control problems. Appl Math Optim 21:53-68 · Zbl 0682.49030 · doi:10.1007/BF01445157
[3] Alt W (1991) Parametric optimization with applications to optimal control and sequential quadratic programming. Bayreuth Math Sehr 35:1-37 · Zbl 0734.90094
[4] Dontchev AL, Hager WW (1993) Lipschitzian stability in nonlinear control and optimization. SIAM J Control Optim 31:569-603 · Zbl 0779.49032 · doi:10.1137/0331026
[5] Dontchev AL, Hager WW (1994) Implicit functions, Lipschitz maps, and stability in optimization. Math Oper Res 19:753-768 · Zbl 0835.49019 · doi:10.1287/moor.19.3.753
[6] Dunn JC, Tian T (1992) Variants of the Kuhn-Tucker sufficient conditions in cones of nonnegative functions. SIAM J Control Optim 30:1361-1384 · Zbl 0768.49015 · doi:10.1137/0330072
[7] Hager WW (1985) Approximations to the multiplier method. SIAM J Numer Anal 22:16-46 · Zbl 0572.65052 · doi:10.1137/0722002
[8] Hager WW (1990) Multiplier methods for nonlinear optimal control. SIAM J Numer Anal 27:1061-1080 · Zbl 0717.49024 · doi:10.1137/0727063
[9] Hager WW, Ianculescu G (1984) Dual approximations in optimal control. SIAM J Control Optim 22:423-465 · Zbl 0555.49022 · doi:10.1137/0322027
[10] Ito K, Kunisch K (1992) Sensitivity analysis of solutions to optimization problems in Hilbert spaces with applications to optimal control and estimation. J Differential Equations 99:1-40 · Zbl 0790.49028 · doi:10.1016/0022-0396(92)90133-8
[11] Malanowski K (1988) On stability of solutions to constrained optimal control problems for systems with control appearing linearly. Arch Automat Telemech 33:483-497 · Zbl 0686.49009
[12] Malanowski K (1992) Second-order conditions and constraint qualifications in stability and sensitivity analysis of solutions to optimizations problems in Hubert spaces. Appl Math Optim 25:51-79 · Zbl 0756.90093 · doi:10.1007/BF01184156
[13] Malanowski K (preprint) Two norm approach in stability and sensitivity analysis of optimization and optimal control problems
[14] Maurer H (1981) First- and second-order sufficient optimality conditions in mathematical programming and optimal control. Math Programming Stud 14:163-177 · Zbl 0448.90069
[15] Maurer H, Zowe J (1979) First- and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems. Math Programming 16:98-110 · Zbl 0398.90109 · doi:10.1007/BF01582096
[16] Orrell K, Zeidan V (1988) Another Jacobi sufficient criterion for optimal control with smooth constraints. J Optim Theory Appl 58:283-300 · Zbl 0629.49016 · doi:10.1007/BF00939686
[17] Reid WT (1972) Riccati Differential Equations. Academic Press, New York
[18] Robinson SM (1976) Stability theory for systems of inequalities, Part II: Differentiable nonlinear systems. SIAM J Numer Anal 13:497-513 · Zbl 0347.90050 · doi:10.1137/0713043
[19] Robinson SM (1980) Strongly regular generalized equations. Math Oper Res 5:43-62 · Zbl 0437.90094 · doi:10.1287/moor.5.1.43
[20] Rudin W (1966) Real and Complex Analysis. McGraw-Hill, New York · Zbl 0142.01701
[21] Ursescu C (1975) Multifunctions with closed convex graph. Czechoslovak Math J 25:438-441 · Zbl 0318.46006
[22] Yang B (1991) Numerical Methods for Nonlinear Optimal Control Problems with Equality Control Constraints. PhD dissertation, Department of Mathematics, Colorado State University, Fort Collins, CO
[23] Zeidan V (1984) Extended Jacobi sufficiency criterion for optimal control. SIAM J Control Optim 22:294-301 · Zbl 0535.49014 · doi:10.1137/0322020
[24] Zeidan V (1984) First- and second-order sufficient conditions for optimal control and the calculus of variations. Appl Math Optim 11:209-226 · Zbl 0558.49006 · doi:10.1007/BF01442179
[25] Zeidan V (1993) Sufficient conditions for variational problems with variable endpoints: coupled points. Appl Math Optim 27:191-209 · Zbl 0805.49012 · doi:10.1007/BF01195982
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