Infinite horizon boundary value problems and applications (Q1300099)

The system of nonlinear differential equations \[ \frac{dX(t)}{dt}=f(t,X(t),Y(t)),\quad X(0)=x_0,\quad \frac{dY(t)}{dt}=g(t,X(t),Y(t)), \] is considered, where \(A=(f,g)\) satisfies the following monotonicity condition: There is a \(\mu>0\) such that \(\langle A(t,\zeta)-A(t,\overline \zeta),\zeta-\overline\zeta\rangle \leq-\mu|\zeta-\overline\zeta|\) for all \(\zeta,\overline\zeta\in \mathbb{R}^{2n}\) and \(t\in \mathbb{R}^+\). It is shown that the problem has a unique solution in \(L^2(0,\infty;\mathbb{R}^n)\) if \(A\) additionally satisfies a uniform Lipschitz condition. Also, asymptotic stability is investigated. An application to the boundary layer terms in the optimal control of a singularly perturbed dynamic system is given.