Dualization and discretization of linear-quadratic control problems with bang–bang solutions

Walter Alt (), C. Yalçın Kaya () and Christopher Schneider ()
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Walter Alt: Friedrich-Schiller-Universität
C. Yalçın Kaya: University of South Australia
Christopher Schneider: Friedrich-Schiller-Universität

EURO Journal on Computational Optimization, 2016, vol. 4, issue 1, No 4, 47-77

Abstract: Abstract We consider linear-quadratic (LQ) control problems, where the control variable appears linearly and is box-constrained. It is well-known that these problems exhibit bang–bang and singular solutions. We assume that the solution is of bang–bang type, which is computationally challenging to obtain. We employ a quadratic regularization of the LQ control problem by embedding the $$L^2$$ L 2 -norm of the control variable into the cost functional. First, we find a dual problem guided by the methodology of Fenchel duality. Then we prove strong duality and the saddle point property, which together ensure that the primal solution can be recovered from the dual solution. We propose a discretization scheme for the dual problem, under which a diagram depicting the relations between the primal and dual problems and their discretization commutes. The commuting diagram ensures that, given convergence results for the discrete primal variables, discrete dual variables also converge to a solution of the dual problem with a similar error bound. We demonstrate via a simple but illustrative example that significant computational savings can be achieved by solving the dual, rather than the primal, problem.

Keywords: Linear-quadratic control; Bang–bang control; Duality; Regularization; Discretization; 49N10; 49N15; 49M25; 49J30; 49J15 (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s13675-015-0049-4

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