Abstract
We propose and compare three numeric algorithms for optimal control of state-linear impulsive systems. The algorithms rely on the standard transformation of impulsive control problems through the discontinuous time rescaling, and the so-called “feedback”, direct and dual, maximum principles. The feedback maximum principles are variational necessary optimality conditions operating with feedback controls, which are designed through the usual constructions of the Pontryagin’s Maximum Principle (PMP); though these optimality conditions are formulated completely in the formalism of PMP, they essentially strengthen it. All the algorithms are non-local in the sense that they are aimed at improving non-optimal extrema of PMP (local minima), and, therefore, show the potential of global optimization.
Highlights
Our study lays in the vein of works [Dykhta, 2014; Dykhta, 2015], where a new sort of necessary optimality conditions is developed for classical and non-smooth optimal control problems
We attempt to extend the mentioned ideas towards the framework of impulsive control [Arutyunov, Karamzin and Lobo Pereira, 2011; Bressan and Rampazzo, 1994; Dykhta, 1990; Gurman, 1972; Karamzin et al, 2014; Krotov, 1996; Miller, 1996; Rishel, 1965; Warga, 1987]. This area of control theory deals with dynamic systems, whose states are discontinuous, while related extremal problems can not be treated by the tools of classical variational analysis; at the same time, such models have behind them rater solid practical motivation [Dykhta and Samsonyuk, 2000; Miller and Rubinovich, 2003; Zavalishchin and Sesekin, 1997]
Based on the previous theoretical results, we develop numeric algorithms for optimal control, which demonstrate a potential of global optimization techniques
Summary
Our study lays in the vein of (relatively recent) works [Dykhta, 2014; Dykhta, 2015], where a new sort of necessary optimality conditions is developed for classical and non-smooth optimal control problems. We attempt to extend the mentioned ideas towards the framework of impulsive control [Arutyunov, Karamzin and Lobo Pereira, 2011; Bressan and Rampazzo, 1994; Dykhta, 1990; Gurman, 1972; Karamzin et al, 2014; Krotov, 1996; Miller, 1996; Rishel, 1965; Warga, 1987] This area of control theory deals with dynamic systems, whose states are discontinuous, while related extremal problems can not be treated by the tools of classical variational analysis; at the same time, such models have behind them rater solid practical motivation [Dykhta and Samsonyuk, 2000; Miller and Rubinovich, 2003; Zavalishchin and Sesekin, 1997]. A similar approach for a nonlinear pre-discretized impulsive control problem had been presented in [Sorokin and Staritsyn, 2018]
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