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Galerkin approximations for the optimal control of nonlinear delay differential equations. (English) Zbl 1405.49018

Kalise, Dante (ed.) et al., Hamilton-Jacobi-Bellman equations. Numerical methods and applications in optimal control. Based on the workshop “Numerical methods for Hamilton-Jacobi equations in optimal control and related fields”, Linz, Austria, November 21–25, 2016. Berlin: De Gruyter (ISBN 978-3-11-054263-9/hbk; 978-3-11-054359-9/ebook). Radon Series on Computational and Applied Mathematics 21, 61-96 (2018).
Summary: Optimal control problems of nonlinear Delay Differential Equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE’s solution. Optimal controls computed from the Pontryagin Maximum Principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding Ordinary Differential Equation (ODE) systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation
For the entire collection see [Zbl 1398.49002].

MSC:

49M25 Discrete approximations in optimal control
34K07 Theoretical approximation of solutions to functional-differential equations
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