Parametric Sensitivity Analysis of Perturbed PDE Optimal Control Problems with State and Control Constraints

• Published in: Journal of optimization theory and applications Vol. 131; no. 1; pp. 17 - 35
• Main Authors: Büskens, C., Griesse, R.
• Format: Journal Article
• Language: English
• Published: New York, NY Springer 01.10.2006; Springer Nature B.V
• Subjects: Applied mathematics; Applied sciences; Boundary conditions; Burgers equation; Control systems; Differentials; Exact sciences and technology; Mathematical models; Methods; Nonlinear programming; Operational research and scientific management; Operational research. Management science; Optimal control; Optimization; Ordinary differential equations; Partial differential equations; Sensitivity analysis; Studies; Sensitivity analysis; Parametric analysis; Parametric sensitivity; parametric sensitivity analysis; Differential system; Non linear control; control-state constraints; Numerical method; Non linear programming; Burgers equation; Quadratic programming; Real time; Partial differential equation; Optimization; Control constraint; Time dependence; nonlinear programming methods; Discretization; Sequential method; Optimal control; Perturbed optimal control problems; partial differential equations; Time optimal control; State constraint
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-006-9122-8

We study parametric optimal control problems governed by a system of time-dependent partial differential equations (PDE) and subject to additional control and state constraints. An approach is presented to compute the optimal control functions and the so-called sensitivity differentials of the optimal solution with respect to perturbations. This information plays an important role in the analysis of optimal solutions as well as in real-time optimal control. The method of lines is used to transform the perturbed PDE system into a large system of ordinary differential equations. A subsequent discretization then transcribes parametric ODE optimal control problems into perturbed nonlinear programming problems (NLP), which can be solved efficiently by SQP methods. Second-order sufficient conditions can be checked numerically and we propose to apply an NLP-based approach for the robust computation of the sensitivity differentials of the optimal solutions with respect to the perturbation parameters. The numerical method is illustrated by the optimal control and sensitivity analysis of the Burgers equation. [PUBLICATION ABSTRACT]