Optimal control of nonlinear hereditary systems
In this paper a functional space \(\Lambda\) is introduced in order to study hereditary control systems of the form \[ (*)\quad x(t)=\phi (0)+\int^{t}_{a}f(s,x_ s,u_ s)ds,\quad t\in [a,b] \] with the initial condition \(x_ a=\phi\) (here, if x is a function defined in [a,b] and \(s\in [a,b]\), then \(x_ s(\theta)=x(s+\theta)\), for \(\theta\in [-r,0])\). The space \(\Lambda\) is isometrically isomorphic to \(L_{\infty}([-r,0])\times {\mathbb{R}}\). It allows us to consider discontinuous controls; moreover, the point evaluation of functions is continuous in \(\Lambda\). Under suitable assumptions, existence, uniqueness and continuous dependence results for solutions of (*) are obtained. An optimization problem is also considered.
scientific article; zbMATH DE number 4000468