バナッハ空間における最適追従問題

抄録

With the aid of functional analysis, the optimal pursuit problem and the optimal final value control problem in Banach spaces are considered. Many papers treat these problems in the Hilbert space setting and derive an adjoint equation and reduce to the solution of the two point boundary value problem. The authors pose the problems in the Banach spaces such as L^p, 1≤p≤+∞. In L¹ and L^∞ spaces the norm is not Fréchet differentiable, not even Gateaux differentiable and admits one-sided Gateaux derivative only.
A necessary and sufficient condition for an optimal control is derived using one-sided Gateaux derivative. The equations of the optimal condition in L¹, L^p (1<p<+∞) and L^∞ spaces are obtained in the explicit form. And finally several examples are illustrated in the Hilbert space setting in order to make this theory understand with ease. The method is straightforward and thus the optimal control can be computed directly.