Computation of Open-Loop Solutions for Zero-Sum Differential Games by Parameterization

Abstract

Given a dynamical system controlled by two parties aiming at diametrically opposite goals and a initial state, regard the time histories of the controls when both parties do the best they can do. In this paper an algorithm is presented to compute them. Mathematically, this conflict situation can be formulated as Zero-Sum Differential Game with terminal payoff and some terminal (equality and/or inequality) constraintsminumaxvΦxtf(P1)subjecttox˙=fx,u,v,O≤t≤tf,x0=xOu∈Uv∈VΨxtf≥¯¯O Restricting the search for the “optimal” opan-loop controls u¯=u¯t,v=v¯t for given xo to a prespecified class of parameterized control functions u = u(t, α), v = v(t, β), α∈RP,β∈Rq, (e.g. The class of continuous and piecewise linear control functions), the following saddlepoint problem in the parameter-space associated with is obtained:minα∈Amaxβ∈BΦxtf,α,β(P2)subjecttoΨxtf,α,β≥¯¯0where x(tf,α, β) is the value of the solution ofx˙=fx,ut,α,vt,β,x0=xOattimetf>o. A⊂RP,B⊂Rqare to be chosen such that ut,α∈U,vt,β∈v for α∈A,β∈B,O≤t≤tf. The algorithm to solve (p2) uses an implementation of the iterative method with quadratic programming of Han-Wilson-Powell for solving general nonlinear optimization problems doing some iteration steps alternating between a and 3. This procedure is continued until both optimizations are completed and thus ends up with a saddlepoint in the parameter space, at least a local one. The partials of Φxtf,α,β,xtf,α,β with respect to α or β needed for each iteration may be computed by forward differences or by the technique of impulsive response functions using solutions of the adjoint variational equations. To demonstrate the feasibility of the algorithm it is used to operate solutions for the “Homicidal Chauffeur”, a classical pursuit-evasion problem in the plane, which can be solved by hand. A second example will be a rather complex and realistic aircraft pursuit-evasion problem, where an exact analytical solution is not found.