Solving control problems over impulse and Heaviside classes of control functions
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SOLVING CONTROL PROBLEMS OVER IMPULSE AND HEAVISIDE CLASSES OF CONTROL FUNCTIONS 1
UDC 517.977.58
K. R. Aida-zade
Optimal control problems over impulse and Heaviside classes of control functions are investigated. Formulas are obtained for the objective functional gradient with respect to the value and, what is more important, to the time they begin to have an effect. The formulas allow using first-order finite-dimensional optimization methods to solve the problems. The results of numerical experiments are presented. Keywords: impulse control, Heaviside function, piecewise constant control, gradient of a functional. INTRODUCTION In many engineering objects and technological processes, continuous or piecewise-continuous (in time) controls cannot be implemented with adequate accuracy. Therefore, of importance are optimal control problems in classes of control functions that can be implemented accurately and easily from the engineering standpoint [1–7]. We will analyze control problems over the classes of Dirac and Heaviside functions. The impulse control theory is now a developing division of dynamic system optimization. The necessary optimality conditions obtained in [1–4] for various systems and definitions of pulse modes as special cases of generalized functions are inconvenient for numerical calculations. Given the number of pulses, we will derive the necessary optimality conditions and constructive formulas for a functional gradient both with respect to pulse strength (unlike many other studies) and to the time of their impact on the system, which will allow applying first-order optimization methods to solve the problems numerically. Mathematical models of many control processes use Heaviside functions as controls. Clearly, such controls are a special case of piecewise-constant functions; however, controls from the class of Heaviside functions are of independent practical interest since each control often takes a constant value and is switched on only once. We will obtain necessary optimality conditions both with respect to values and, which is most important, with respect to the time of control onset. These conditions contain formulas for the gradient of a functional in the space of parameters being optimized and allow using first-order finite-dimensional optimization methods to solve optimal control problems. Numerical results will be presented. 1. PROBLEM STATEMENT Consider an optimal control problem T
J ( u ) = a1 ò f 0 ( x, t ) dt + a 2 F1 ( x(T )) + a 3 F 2 ( u( t )) ® min ,
(1)
u (t )
0
L
x& i ( t ) = f i ( x( t ), t ) + å bij ( t ) u ij ( t ), 0 < t £ T , x( 0) = x 0 , i = 1, n ,
(2)
j =1
1
The study was supported by the INTAS (grant No. 06-1000017-8909) within the framework of the INTAS program for the promotion of cooperation with scientists from Southern Caucasian Republics, 2006. Institute of Cybernetics, National Academy of Sciences of Azerbaijan, Baku, Azerbaijan, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 137–145, September–October 2009. Origi
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