# American Institute of Mathematical Sciences

September  2015, 35(9): 4367-4384. doi: 10.3934/dcds.2015.35.4367

## Optimal control of dynamical systems with polynomial impulses

 1 Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russian Federation, Russian Federation

Received  April 2014 Revised  September 2014 Published  April 2015

The paper is devoted to the $BV$-relaxation of a dynamical system, whose right-hand side is a $p$th degree polynomial with rational powers of control under a uniform bound on its $L_p$-norm, and coefficients containing usual measurable bounded control.
Under natural convexity assumptions, we give an explicit representation of generalized solutions to the control system by a measure differential equation. The main results concern an optimal impulsive control problem for the relaxed system: We establish the existence of a minimizer, and give necessary optimality conditions in the form of a Maximum Principle.
Citation: Elena Goncharova, Maxim Staritsyn. Optimal control of dynamical systems with polynomial impulses. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4367-4384. doi: 10.3934/dcds.2015.35.4367
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##### References:
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