The main focus of this paper is to develop a sufficiency criterion for optimality in nonlinear optimal control problems defined on time scales. In particular, it is shown that the coercivity of the second variation together with the controllability of the linearized dynamic system are sufficient for the weak local minimality. The method employed is based on a direct approach using the structure of this optimal control problem. The second aim pertains to the sensitivity analysis for parametric control problems defined on time scales with separately varying state endpoints. Assuming a slight strengthening of the sufficiency criterion at a base value of the parameter, the perturbed problem is shown to have a weak local minimum and the corresponding multipliers are shown to be continuously differentiable with respect to the parameter. A link is established between (i) a modification of the shooting method for solving the associated boundary value problem, and (ii) the sufficient conditions involving the coercivity of the accessory problem, as opposed to the Riccati equation, which is also used for this task. This link is new even for the continuous time setting.
Accepted:
DOI: 10.1051/cocv/2017070
Keywords: Optimal control problem on time scales, Weak Pontryagin maximum principle, Weak local minimum, Coercivity, Sufficient optimality condition, Sensitivity analysis, Second variation, Controllability
@article{COCV_2018__24_4_1705_0, author = {\v{S}imon Hilscher, Roman and Zeidan, Vera}, title = {Sufficiency and sensitivity for nonlinear optimal control problems on time scales via coercivity}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1705--1734}, publisher = {EDP-Sciences}, volume = {24}, number = {4}, year = {2018}, doi = {10.1051/cocv/2017070}, mrnumber = {3922438}, zbl = {1415.49015}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017070/} }
TY - JOUR AU - Šimon Hilscher, Roman AU - Zeidan, Vera TI - Sufficiency and sensitivity for nonlinear optimal control problems on time scales via coercivity JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1705 EP - 1734 VL - 24 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017070/ DO - 10.1051/cocv/2017070 LA - en ID - COCV_2018__24_4_1705_0 ER -
%0 Journal Article %A Šimon Hilscher, Roman %A Zeidan, Vera %T Sufficiency and sensitivity for nonlinear optimal control problems on time scales via coercivity %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1705-1734 %V 24 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017070/ %R 10.1051/cocv/2017070 %G en %F COCV_2018__24_4_1705_0
Šimon Hilscher, Roman; Zeidan, Vera. Sufficiency and sensitivity for nonlinear optimal control problems on time scales via coercivity. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 4, pp. 1705-1734. doi : 10.1051/cocv/2017070. http://archive.numdam.org/articles/10.1051/cocv/2017070/
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