Numerical Solution of Minimax Optimal Control Problems by Multiple Shooting

Abstract

This paper presents a numerical method for the solution of optimal control problems of the Čebšev - type : Minimize the functional I [ x,u ] = max 0 ≤ t ≤ t f C ( x ( t ) ) + ∫ 0 t f L ( x, u ) dt subject to the constraints x ˙ = f x, u , 0 ≤ t ≤ t f , x t ɛ R n r x 0 , x t f = 0 ɛ R k u t ɛ U C R 1 The genera] way to attack optimal control problems by indirect methods is to derive necessary conditions in order to build up a boundary value problem with respect to the state variables and the aajoints. For this purpose a standard - transformation due to Warga has been used. This technique converts the minimax problem into an equivalent optimal control problem with a state — variable inequality constraint. Using this way the highly developed theory on the necessary conditions for state — restricted problems can be applied advantageously. In addition the results concerning the order of the constraint are transfered to minimax problems. The cases of regular and singular Hamiltonian are to be distinguished. In the regular case the necessary conditions mainly coincide with the conditions earlier derived by Powers. In addition, however, one obtains a sign — condition for the Lagrange multiplier corresponding to the state constraint. More complicated is the singular case, i.e. the problem depends linearly on the control variable. This is due to the more complicated switching structure of these problems. The junction conditions for state restricted problems developed by Maurer are transfered to the minimax case. These conditions are most important in order to predict whether the maximum of the Čebyšev — function C(x(t)) is unique or flat. In particular, it is shown that a singular problem of the order one may have a unique maximum. Evidently, this is impossible in the regular case. Assuming that the switching structure of the solution trajectory can be estimated, a boundary value problem for the state and adjoint variables is derived. This boundary value problem contains certain boundary conditions which are to be satisfied at (unknown) switching points. Such problems can be treated numerically by the earlier developed multiple shooting code BOUNDSCO. This routine is briefly described. It is a modification of the well known multiple shooting code BOUNDSOL for two—point boundary value problems due to Bulirsch. The method is applied to two real-life problems. The first example is the well-known reentry problem. The pay-off is changed in order to minimize a combination of the total stagnation point convective heating rate and the maximal heating rate : I = c 1 max 0 ≤ t ≤ t f q ˙ t + C 2 ∫ 0 t f q ˙ t dt where q ˙ = 10 v 3 ζ v: velocity, ζ: atmospheric density. This regular minimax problem is of the order one. Therefore the optimal heating rate q ˙ (t) has a flat maximum. Numerical solutions of this problem are obtained by a homotopy method with respect to the weights C 1 and C 2 . The second example is a realistic model for optimal heating and cooling of a house by solar energy. The aim is to determine control histories for heating and cooling such that the inside temperature has a minimal deviation from a desired value T D : I = c 1 max 0 ≤ t ≤ t f T I − T D + C 2 ∫ 0 24 T I − T D 2 dt. The resulting minimax problem is singular and of the order one. For various values of C 1 and C 2 numerical solutions of this problem are presented.