Optimal control problems over large time intervals

The paper focuses on problems af optimal control over a large time interval with end states prescribed. For a wide class of systems and performance indices, an approximate solution is characterized by piecing together two solutions for two infinite time problems. These solut~ons exhibit similar transient behaviour. L INTRODUCTION CONSIDER the following linear quadratic optimal o T control problem: for the time-invariant, completely ....". controllable system FIG. 1. Optimal trajectory. with prescribed x(O), x(T), choose u(.) to minimize the performance index J(x(O), ti(.), T) = [u'Ru + x'Qx] dt (1.2) 1 with R > 0, Q > 0 and where [F, Q1''] is completely observable. When T becomes large, the trajectories acquire several semiquantitative properties: (i) most of the transient activity occurs near t = 0 and t = T; (ii) x(t) remains very small away from t = 0 and t = T; *Received 29 May 1986; revised 9 October 1986. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor P. Guimaraes Ferreira under the direction of Editor H. Kwakernaak. tDepartment of Systems Engineering, Research School of Physical Sciences, Australian National University, P.O. Box 4, Canberra, A.C.T., 2601, Australia. To whom correspondence should be addressed. f Department of Electrical Engineering, University of Illinois, Urbana, Illinois, U.S.A.; formerly Visiting Fellow, Department of Systems Engineering, Research School of Physical Sciences, Australian National University, G.P.O. Box 4, Canberra, A.C.T., 2601, Australia. (iii) the time constants associated with the trajectory near t = 0 are the same as those associated with the trajectory near t = T; (iv) the optimal trajectory and control for the original problem can be approximately obtained by piecing together the optimal trajectory and control associated with the two indices with x(0) prescribed (the solution defines the part of the trajectory for the original problem near x = 0) and with x(T) prescribed, the value of x(-m) being immaterial. (Here, the solution defines the part of the trajectory of the original problem near t = T ) Properties (i)-(iii) are illustrated in Fig. 1 and property (iv) by Figs 1 and 2. For a precise formulation of the basic ideas, see Wilde and Kokotovic (1972), and for some extensions and additional remarks see Wilde and Kokotovic (1973), Kokotovic et al. (1976), Kokotovic (1984) and