Abstract
We develop a theory for sensitivity with respect to parameters in a convex subset of a topological vector space of dynamical systems in a Banach space. Specific motivating examples for probability measure dependent differential, partial differential and delay differential equations are given. Schemes that approximate the measures in the Prohorov sense are illustrated with numerical simulations for distributed delay differential equations.
Highlights
IntroductionWhose solution x is assumed to exist and belong to a complex Banach space (X, | · |X) for any choice of the parameter μ in a convex subset M of a topological vector space X
In this paper we develop sensitivity equations for a general nonlinear dynamical system in a Banach space depending on parameters in a convex subset of a topological vector space
In this paper we have developed both theoretical and computational aspects of a sensitivity methodology for probability measure dependent dynamical systems
Summary
Whose solution x is assumed to exist and belong to a complex Banach space (X, | · |X) for any choice of the parameter μ in a convex subset M of a topological vector space X. S0(x) denotes the initial population density of susceptible shrimp produced from the biomass production model We note that this is again a probability distribution (PL) dependent dynamical system (in this case a complicated system of partial differential equations) for which the distribution PL must be estimated in some type of inverse problem. Special cases of such systems include those in which the probability measures are defined on some finite interval Q = [−r, 0] of possible delay values θ In this model one has compartments T, A, C, and V for in vitro blood level counts in mice of target (CD4+) cells, acutely infected cells, chronically infected cells and active viral particles, respectively. Our group is currently using models such as (20) with the data investigated in [3]; initial results are quite promising
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