Integral Manifold and Weak Nonlinear Control System

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 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.

Integral manifold andH? control in weak nonlinear systems

“…A brief outline of derivation of the above inequalities duplicates the proof of a theorem disclosed in Ref. 6. Therefore, see Appendix 3.…”

“…0, x 0 , q, ht qxt; 0, x 0 , q in place of xt, qt in the function F, we have This Tqx 0 is a function of only the initial point x 0 and the function q; thus, it is a constant regardless of the initial moment [6].…”

“…In addition, based on the discussion in Ref. 6, lt can be obtained as the state feedback C CH qx t depending on x t, using the function qx. Therefore, by solving the system of differential equations x t, lt using Eqs.…”