Abstract
In this paper, we use the fixed point theorem to generate a set of sufficient conditions which confine the solutions of weak non-linear differential equations to a stable integral manifold. The obtained conditions are used in the solution of a weak non-linear optimal regulator problem. The necessary conditions for optimal control laws are expressed by the weak non-linear differential equations with state variable x and Lagrange's variable λ. By using the solution S of the Ricatti equation, the sufficient conditions for obtaining the solution λ(t) in a feedback form, λ(t)=Sx(t)+p(x(t)), are derived from the result of the integral manifold. The method for finding an approximate solution of p(x) in an analyltic form such as a Fourier series is introduced, and a new calculation algorithm is presented which applies the Newton method and other techniques. Simulations using the algorithm exhibit favorable results.
