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Some results on optimal control with unilateral state constraints. (English) Zbl 1165.49036
Summary: We study the problem of quadratic optimal control with state variables unilateral constraints, for linear time-invariant systems. The necessary conditions are formulated as a linear invariant system with complementary slackness conditions. Some structural properties of this system are examined. Then it is shown that the problem can benefit from the higher order Moreau’s sweeping process, that is, a specific distributional differential inclusion, and from ten Dam’s geometric theory [A. A. ten Dam, K. F. Dwarshuis and J. C. Willems, IEEE Trans. Autom. Control 42, No. 4, 458–472 (1997; Zbl 0886.93031); A. A. ten Dam, Unilaterally constrained dynamical systems, Ph.D. Thesis, Rijsuniversiteit Groningen, NL, available at http://irs.ub.rug.nl/ppn/159407869 (1997)] for partitioning of the admissible domain boundary (in particular for the case of multivariable systems). In fact, the first step may be also seen as follows: does the higher order Moreau’s sweeping process (developed in [V. Acary, B. Brogliato and D. Goeleven, Math. Program. 113, No. 1 (A), 133–217 (2008; Zbl 1148.93003)]) correspond to the necessary conditions of some optimal control problem with an extended integral action? The knowledge of the qualitative behaviour of optimal trajectories at junction times is improved with the approach, which also paves the way towards efficient time-stepping numerical algorithms to solve the optimal control boundary value problem.

MSC:
49N10 Linear-quadratic optimal control problems
49K15 Optimality conditions for problems involving ordinary differential equations
Software:
BNDSCO
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