An approximation to a differential inclusion is presented in the linear case. Differential inclusions of the form (1) \(\mathring x\in A(t)x+B(t)U\), \(x(t_ 0)\in X_ 0\subset\mathbb{R}^ n\) are considered, where \(A(\cdot)\) and \(B(\cdot)\) are differentiable with Lipschitz continuous derivatives and \(X_ 0\) and \(U\subset\mathbb{R}^ r\) are convex and compact.
A discrete recurrent formula of the form \(X^ N_{k+1}=A_ h(k)X^ N_ k+B_ h(k)U\), \(k=0,\dots,N-1\), \(X^ N_ 0=X_ 0\) is defined and it is shown that it provides a second order approximation to (1) i.e. \(H(X^ N_ N,R(X_ 0;t_ 0,T))\leq \text{const}\cdot h^ 2\), where \(H\) is the Hausdorff metric and \(R(X_ 0;t_ 0,T)\) is the reachable set of (1) on \([t_ 0,T]\) starting from \(X_ 0\).
This result is applied to obtain a second-order discrete approximation to a Lagrange optimal control problem by means of a sequence of mathematical programming problems. A Hamilton-Jacobi theory for this Lagrange problem is discussed and an approximate feedback synthesis is derived using the above second order approximation, which ensures \(O(1/N^ 2)\)- approximation to the optimal value.
Finally, it is shown that better approximations than of second order of accuracy cannot be obtained by means of this type of discretizations.