In this paper we develop a new approach for optimal control problems with general jointly varying state endpoints (also called coupled endpoints). We present a new transformation of a nonlinear optimal control problem with jointly varying state endpoints and pointwise equality control constraints into an equivalent optimal control problem of the same type but with separately varying state endpoints in double dimension. Our new transformation preserves among other properties the controllability (normality) of the considered optimal control problems. At the same time it is well suited even for the calculus of variations problems with joint state endpoints, as well as for optimal control problems with free initial and/or final time. This work is motivated by the results on the second order Sturm–Liouville eigenvalue problems with joint endpoints by Dwyer and Zettl (1994) and by the sensitivity result for nonlinear optimal control problems with separated state endpoints by the authors (2018).
Accepté le :
Première publication :
Publié le :
Keywords: Optimal control problem, joint (coupled) endpoints, separated endpoints, controllability, strong Pontryagin principle, coercivity, sensitivity analysis, free time problem
@article{COCV_2021__27_1_A77_0,
author = {\v{S}imon Hilscher, Roman and Zeidan, Vera M.},
title = {Transformation preserving controllability for nonlinear optimal control problems with joint boundary conditions},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021068},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021068/}
}
TY - JOUR AU - Šimon Hilscher, Roman AU - Zeidan, Vera M. TI - Transformation preserving controllability for nonlinear optimal control problems with joint boundary conditions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021068/ DO - 10.1051/cocv/2021068 LA - en ID - COCV_2021__27_1_A77_0 ER -
%0 Journal Article %A Šimon Hilscher, Roman %A Zeidan, Vera M. %T Transformation preserving controllability for nonlinear optimal control problems with joint boundary conditions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021068/ %R 10.1051/cocv/2021068 %G en %F COCV_2021__27_1_A77_0
Šimon Hilscher, Roman; Zeidan, Vera M. Transformation preserving controllability for nonlinear optimal control problems with joint boundary conditions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 75. doi: 10.1051/cocv/2021068
[1] and , Second order conditions for periodic optimal control problems. Control Cybernet. 34 (2005) 617–643.
[2] , , and , Quadratic order conditions for bang-singular extremals Numer. Algebra Control Optim. 2 (2012) 511–546.
[3] , Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions. J. Math. Anal. Appl. 199 (1996) 804–826.
[4] , Sufficient conditions for optimality and the justification of the dynamic programming method. SIAM J. Control Optim. 4 (1966) 326–361.
[5] , and , Second-order necessary conditions in Pontryagin form for optimal control problems. SIAM J. Control Optim. 52 (2014) 3887–3916.
[6] and , Pontryagin maximum principle for finite dimensional nonlinear optimal control problems on time scales. SIAM J. Control Optim. 51 (1996) 3781–3813.
[7] and , Optimal sampled-data control, and generalizations on time scales. Math. Control. Relat. Fields 6 (1996) 53–94.
[8] , Eigenvalues of Matrix Sturm–Liouville Problems with Separated or Coupled Boundary Conditions. Ph.D. dissertation, Northern Illinois University, DeKalb, IL (1993).
[9] and , Computing eigenvalues of regular Sturm–Liouville problems. Electr. J. Differ. Equ. 1994 (1996) 10 pp.
[10] and , Discrete oscillation theorems for symplectic eigenvalue problems with general boundary conditions depending nonlinearly on spectral parameter. Linear Algebra Appl. 558 (1996) 108–145.
[11] and , Deterministic and Stochastic Optimal Control, Applications of Mathematics, No. 1. Springer-Verlag, Berlin-New York (1996).
[12] and , Riccati inequality and other results for discrete symplectic systems. J. Math. Anal. Appl. 322 (1996) 1083–1098.
[13] and , Symplectic difference systems: variable stepsize discretization and discrete quadratic functionals. Linear Algebra Appl. 367 (1996) 67–104.
[14] and , Applications of time scale symplectic systems without normality. J. Math. Anal. Appl. 340 (1996) 451–465.
[15] and , Time scale embedding theorem and coercivity of quadratic functionals. Analysis (Munich) 28 (1996) 1–28.
[16] and , Riccati equations for abnormal time scale quadratic functionals. J. Differ. Equ. 244 (1996) 1410–1447.
[17] and , Weak maximum principle and accessory problem for control problems on time scales. Nonlinear Anal. 70 (1996) 3209–3226.
[18] , Extensions of Linear-Quadratic Controls, Optimization and Matrix Theory, Vol. 133 of Mathematics in Science and Engineering. Academic Press, London – New York – San Francisco (1996).
[19] , The theory of optimal control and the calculus of variations. In Mathematical Optimization Techniques”, Symposium on Mathematical Optimization Techniques, Santa Monica, CA, 1960, edited by . Univ. California Press, Berkeley, CA (1996) 309–331.
[20] , Quadratic Functionals in Variational Analysis and Control Theory, Vol. 6 of Mathematical Topics. Akademie Verlag, Berlin (1996).
[21] , Linear and Nonlinear Programming, Addison-Wesley, Reading, MA (1996) 2nd edn.
[22] and , Sensitivity analysis for parametric control problems with control-state constraints. Comput. Optim. Appl. 5 (1996) 253–283.
[23] , Sufficient conditions and sensitivity analysis for economic control problems in “Optimal Control and Differential Games” (Vienna, 1997). Ann. Oper. Res. 88 (1996) 3–14.
[24] and , Solution differentiability for nonlinear parametric control problem. SIAM J. Control Optim. 32 (1996) 1542–1554.
[25] and , Solution differentiability for parametric nonlinear control problems with control-state constraints. J. Optim. Theory Appl. 86 (1996) 285–309.
[26] and , Second-order sufficient conditions for control problems with mixed control-state constraints. J. Optim. Theory Appl. 86 (1996) 649–667.
[27] and , Another Jacobi sufficiency criterion for optimal control with smooth constraints. J. Optim. Theory Appl. 58 (1996) 283–300.
[28] , Sufficient quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints. J. Math. Sci. (N.Y.) 173 (1996) 1–106.
[29] and , Bang-singular-bang extremals: sufficient optimality conditions. J. Dyn. Control Syst. 17 (1996) 469–514.
[30] , Eigenvalue theory for time scale symplectic systems depending nonlinearly on spectral parameter. Appl. Math. Comput. 219 (1996) 2839–2860.
[31] and , Hamilton–Jacobi theory over time scales and applications to linear-quadratic problems. Nonlinear Anal. 75 (1996) 932–950.
[32] and , Oscillation theorems and Rayleigh principle for linear Hamiltonian and symplectic systems with general boundary conditions. Appl. Math. Comput. 218 (1996) 8309–8328.
[33] and , Sufficiency and sensitivity for nonlinear optimal control problems on time scales via coercivity. ESAIM: COCV 24 (1996) 1705–1734.
[34] and , Weyl disks and square summable solutions for discrete symplectic systems with jointly varying endpoints. Adv. Differ. Equ. 2013 (1996) 232..
[35] and , Primer on Optimal Control Theory, Vol. 20 of Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2010).
[36] and , Constrained regular LQ-control problems. SIAM J. Control Optim. 35 (1996) 876–900.
[37] , Optimal Control, Systems & Control: Foundations & Applications, Birkhäuser, Boston, MA (1996).
[38] , Sufficiency conditions with minimal regularity assumptions. Appl. Math. Optim. 20 (1996) 19–31.
[39] , The Riccati equation for optimal control problems with mixed state-control constraints: necessity and sufficiency. SIAM J. Control Optim. 32 (1996) 1297–1321.
[40] , New second-order optimality conditions for variational problems with C2-Hamiltonians. SIAM J. Control Optim. 40 (1996) 577–609.
[41] and , Normality for the problem of Bolza with an inequality state constraint Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology (Edmonton, AB, 1988). Rocky Mountain J. Math. 20 (1996) 1235–1248.
Cité par Sources :
This research was supported by the Czech Science Foundation under grant GA19–01246S.
