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This paper discusses existence, uniqueness, and a priori estimates for time-dependent and time-independent transport equations with unbounded collision operators. These collision operators are described by second-order differential operators resulting from diffusion in the velocity space. The transport equations are degenerate parabolic-elliptic partial differential equations, that are treated by modifications of the Fichera–Oleinik–Radkevic Theory of second-order equations with nonnegative characteristic form. We consider weak solutions in spaces that are extensions of $L^P $ to include traces on certain parts of the boundary. This extension is necessary due to the nonclassical boundary conditions imposed by the transport problem, which requires a specific analysis of the behavior of our weak solutions.

On etablit des conditions necessaires et suffisantes pour qu'une classe d'equations aux derivees partielles fonctionnelles soit asymptotiquement stable pour toutes les valeurs des retards

This paper concerns the optimal stopping time problem for a piecewise deterministic process. The process has deterministic dynamics between random jumps. The as¬sociated dynamic programming equation is a variational inequality with integral and (first order) differential terms. Our main results are W4,00-existence results and probabilistic representations for the solutions of the optimal stopping time problem in bounded domains and in R. We also generalize these results to the case when the state space is “countable folds” of Euclidean space

The shape and propagation of waves produced by eukaryotic flagella depend on the three-dimensional arrangement and physical-chemical properties of peripheral substructures. The modeling analysis presented here, which assumes force-moment equilibrium and neglects the viscous resistances of the medium, shows how substructural arrangements characteristic of 9+0, 9+1, and 9+2 axonemes can yield their characteristic wave patterns. When flexural stiffnesses are equal along all axonemal radii, any non-uniform doublet shearing pattern propagated distally at constant rate, with successive pairs 1 9 cycle out of phase, should generate helical waves. When stiffnesses differ greatly on different radii, but the stiffness pattern is the same for all cross-sections, any such shearing pattern should yield planar waves resembling sine-generated curves. Propagated axonemal bending results from the active bending moment produced by local shearing of doublet pairs. Uniformly twisting the doublets about the axonemal axis cannot directly alter the magnitude of the active bending moment. If dynein cross-bridges are activated by shear displacement between peripheral doublets, then the resulting distribution of the active bending moment will be appropriate for balancing the elastic moment in a propagated bending wave.

We consider viscosity solutions for weakly coupled systems of first-order partial differential equations and apply these existence and uniqueness results to examples involving random evolutions.

We show existence, uniqueness, andW 2,∞ -regularity of the system of nonlinear partial differential equations, associated with stochastic optimal control problems involving costly switchings and impulses. The impulse obstacle is approximated by a sequence of switching obstacles in the proof of regularity. The semiconcavity of the impulse obstacle is exploited.

We present a model of infectious diseases in heterogeneous populations, which allows for variable intra- to intergroup contact ratios. We give necessary and sufficient conditions for disease eradication by means of vaccination. Smallpox is used as an illustrative example.

A finite collection of piecewise-deterministic processes are controlled in order to minimize the expected value of a performance functional with continuous operating cost and discrete switching control costs. The solution of the associated dynamic programming equation is obtained by an iterative approximation using optimal stopping time problems.

We consider a bilinear optimal control problem where the state equation is a Kirchhoff plate equation. The control acts as a multiplier of the velocity term. We prove the existence of an optimal control in a class h ∈ UM = {h ∈ L∞(0, T); −M ≤ h(t) ≤ M} and uniqueness of this optimal control for T sufficiently small.

This paper concerns the optimal stopping time problem for a piecewise deterministic process. The associated dynamic programming equation is a variational inequality with integral and first order differential terms Our main results are W1,?-existence results and probabilistic representations for the solutions of these variational inequalities in bounded domains and in IRn.

Existence and regularity results are obtained for Bellman equation with integro-differential operators, corresponding to the stochastic control of diffusion processes with jumps.

We study the problem of optimal control of a deterministic system with final state constraints using dynamic programming methods. We present a penalization scheme and regularity results for the optimal cost function.

We consider switching games for piecewise-deterministic processes and the associated system of integro-differential equations. The viscosity solution of the system on Rn is a saddle point for the game.

This paper considers existence and uniqueness results for viscosity solutions of integro-differential equations associated with the impulse control problem for piecewise-deterministic processes on bounded domains and on Rn.

We study the problems of optimal switching control of diffusion processes with additional obstacles and switching control of random evolutions.

Using an existing ordinary differential equation model which describes the interaction of the immune system with the human immunodeficiency virus (HIV), we introduce chemotherapy in an early treatment setting through a dynamic treatment and then solve for an optimal chemotherapy strategy. The control represents the percentage of effect the chemotherapy has on the viral production. Using an objective function based on a combination of maximizing benefits based on T cell counts and minimizing the systemic cost of chemotherapy (based on high drug dose/strength), we solve for the optimal control in the optimality system composed of four ordinary differential equations and four adjoint ordinary differential equations.

We consider certain constrained controllability and approximate controllability properties of a non-linear system that can be deduced from various controllability properties of its associated linear system.

Considers the problem of controlling the solution of a Kirchhoff plate equation. The equation with appropriate boundary conditions describes the motion of a thin plate which is clamped along one portion of its boundary and has free vibrations on the other portion of its boundary. The authors consider bilinear optimal control in which the control acts as a distributed force. >

An SIR type model is expanded to include the use of education or information given to the public as a control to manage a disease outbreak when effective treatments or vaccines are not readily available or too costly to be widely used. The information causes a change in behavior resulting in three susceptible classes. We study stability analysis and use optimal control theory on the system of differential equations to achieve the goal of minimizing the infected population (while minimizing the cost). We illustrate our results with some numerical simulations.

A two-sided game for the control of a stationary semilinear competitive system with non-local interaction terms is considered. The controls of the system are the autonomous sources. The saddle point of the system, i.e. the optimal solution of the game, is characterized as the unique solution of the associated optimality system (OS). The OS consists of two elliptic partial differential equation state equations coupled with two adjoint equations. The solution of the OS is constructed by an alternating monotonic iteration scheme. >

A finite collection of piecewise-deterministic processes are controlled in order to minimize the expected value of a performance functional with continuous and switching control costs. The solution of the associated dynamic programming equation is obtained by an iterative approximation using optimal stopping time problems.

Optimal control theory is applied to a system of ordinary differential equations modeling a two-strain tuberculosis model. Seeking to reduce the latent and infectious groups with the resistant-strain tuberculosis, we use controls representing two types of treatments. The optimal controls are characterized in terms of the optimality system, which is solved numerically for several scenarios.

The common assumption in much of the literature that an asymptotically stable system will result in bounded covariance in the limit as t ? ? is disproved by a simple counter-example. Necessary and sufficient conditions for bounded covariance are given and proved.