On the set-valued calculus in problems of viability and control for dynamic processes: The evolution equation
This paper introduces an evolution equation for the tube \(X(t,t_ 0,X^ 0)\) of all the solutions to a differential inclusion \(\dot x\in F(t,x)\), \(x^ 0\in X^ 0\) that also satisfy a state constraint \(x\in Y(t)\), \(t_ 0\leq t\leq t_ 1\). Here \(F(t,x)\), \(Y(t)\) are convex compact and closed convex-valued, continuous in x, t and in t respectively. Under some additional assumptions it is indicated that the multivalued map \(X[t]=X(t,t_ 0,X^ 0)\), \(X[t_ 0]=X^ 0\) satisfies the evolution ``funnel'' equation \[ \lim_{\sigma \to +0}\sigma^{-1} h(X[t+\sigma],\cup_{x\in X[t]}(x+\sigma F(t,x))\cap Y(t+\sigma))=0, \] where \(h(X',X'')\) is the Hausdorff distance between \(h(X',X'')\). A proof of the latter statement is followed by another ``direct'' proof for the case when the map \(F(t,x)=A(t)x+P(t)\) is linear in x (P(t) being a multivalued map). A set-valued Lagrangian technique is then introduced for direct calculation of the support function \(\rho (\ell | X[t])=\max \{(\ell,x)|\) \(x\in X[t]\}\) and its evolution in time t. It is also shown that for the latter case the set \(X[t]=\{\cap X(t;L(\cdot))| L(\cdot)\}\) over all continuous matrix functions L(t), where X(t;L(\(\cdot))\) is the tube of solutions of a differential inclusion \(\dot x(t)\in (A(t)-L(t))x+P+L(t)Y(t)\) with some initial set \(X[t_ 0]=X^ 0\) as above. A directional derivative of type (\(\partial /\partial t)\rho (\ell | X[t])\) is further calculated and the explicit formula obtained here is then used for the proof of the existence of a multivalued synthesizing strategy U(t,x) that steers the system \(\dot x\in A(t)x+U(t,x)\) from a preassigned position \((r,x^*)\) to a given position \((t_ 1,z)\), \(z\in M\) (M is a given convex target set in \({\mathbb{R}}^ n)\), under the restrictions \(x(t)\in Y(t)\), \(U(t,x)\subseteq P(t)\).
scientific article; zbMATH DE number 4134914