Sensitivity analysis of trajectories on infinite time horizon
Abstract
A central problem in the theory and applications of differential inclusions is the continuous dependence of solutions on the initial conditions.
Several papers have studied this problem, and it seems that basically there are two major approaches. One of them is establishing the existence of a continuous map
$\xi \mapsto x_\xi $} such that $x_\xi $}
is the Caratheodory-solution to the Cauchy-problem (1).
The other one is proving the continuity of the set valued map
$\xi \mapsto S(\xi) $, where $ S(\xi) $
denotes the set of solutions starting from $\xi$.
Continuity of the set valued map $ S $} is understood with respect to the Hausdorff-metric induced by the norm of the space solutions.
Here we refer to Lim [9], who used a contraction principle for set valued maps, or to Deimling [5] Lemma 8.3, about the application of the so-called Filippov-Grönwall-inequality. Ultimately, both methods are based on a certain iteration scheme.
In [4] Constantin proved the continuity of the solution map for infinite time horizon problems, by introducing an appropriate norm on the space of continuous functions. His proof relies on stability properties of fixed point sets of set valued contractions obtained by Lim [9].
In the present paper we extend that infinite time horizon result under significantly milder conditions, by observing that much stronger statements can be obtained, if the contraction principle is applied in the space of the derivatives of solutions instead of the space of solutions. In contrast to Constantin [4], we do not need the convexity or the compactness of the right-hand side, nor do we use continuity, only measurability is assumed for $F$ with respect to $t$. Also, the norm we introduce on the space of solutions generates a stronger topology than that of Constantin, so our continuity result is a refinement of the existing theorems. Finally, the comparison of the norms allows us to derive a Filippov-Grönwall-type inequality in infinite dimensional spaces.
