Viability Kernels of Differential Inclusions with Constraints: Algorithms and Applications

The authors investigate a differential inclusion whose solutions have to remain in a given closed set. The viability kernel is the set of the initial conditions starting at which, there exist solutions to the differential inclusion remaining in this closed set. In this paper, the authors provide an algorithm which determine this set and they apply it to some concrete examples

Dissipative Control Systems and Disturbance Attenuation for Nonlinear H - Problems

We characterize functions satisfying a dissipative inequality associated with a control problem. Such a characterization is provided in terms of epicontingent and viscosity supersolutions to a Partial Differential Equation called the Hamilton-Jacobi-Bellman-Isaacs equation. Links between viscosity and epicontingent supersolutions are studied. Finally, we derive (possibly discontinuous) disturbance attenuation feedback of the H^{infty}-problem from contingent formulation of the Isaacs' Equation

Isaacs' Equations for Value-Functions of Differential Games

The authors study value functions of a differential game with payoff which depends on the state at a given end time. They consider differential games with feedback strategies and with nonanticipating strategies. They prove that value-functions are solutions to some Hamilton-Jacobi-Isaacs equations in the viscosity and contingent sense. For these two notions of strategies, with some regularity assumptions, The authors prove that value-functions are the unique solution of Isaacs' equations

On second order necessary conditions in infinite dimensional optimal control with state constraints

This paper is devoted to second order necessary optimality conditions for control problems in infinite dimensions. The main novelty of our work is the presence of pure state constraints together with end point constraints, quite useful in the applications. Second order analysis for control problems involving PDEs has been extensively discussed in the literature. The most usual approach to derive necessary optimality conditions is to rewrite the control problem as an abstract mathematical programming one. Our approach is different, we avoid the reformulation of the optimal control problem and use instead second order variational analysis. The necessary optimality conditions are in the form of a maximum principle and a second order variational inequality. They are first obtained in the form of nonintersection of convex sets. A suitable separation theorem allows to deduce their dual characterization

Cocaine Self-Administration and Abstinence Modulate NMDA Receptor Subunits and Active Zone Proteins in the Rat Nucleus Accumbens

Cocaine-induced plasticity in the glutamatergic transmission and its N-methyl-d-aspartate (NMDA) receptors are critically involved in the development of substance use disorder. The presynaptic active zone proteins control structural synaptic plasticity; however, we are still far from understanding the molecular determinants important for cocaine seeking behavior. The aim of this study was to investigate the effect of cocaine self-administration and different conditions of cocaine forced abstinence on the composition of the NMDA receptor subunits and on the levels of active zone proteins, i.e., Ras-related protein 3A (Rab3A), Rab3 interacting molecules 1 (RIM1) and mammalian uncoordinated protein 13 (Munc13) in the rat nucleus accumbens. We found an up-regulation of the accumbal levels of GluN1 and GluN2A following cocaine self-administration that was paralleled by an increase of Munc13 and RIM1 levels. At the same time, we also demonstrated that different conditions of cocaine abstinence abolished changes in NMDA receptor subunits (except for higher GluN1 levels after cocaine abstinence with extinction training), while an increase in the Munc13 concentration was shown in rats housed in an enriched environment. In conclusion, cocaine self-administration is associated with the specific up-regulation of the NMDA receptor subunit composition and is related with new presynaptic targets controlling neurotransmitter release. Moreover, changes observed in cocaine abstinence with extinction training and in an enriched environment in the levels of NMDA receptor subunit and in the active zone protein, respectively, may represent a potential regulatory step in cocaine-seeking behavior

Proteolysis dysfunction in the process of aging and age-related diseases

In this review, we discuss in detail the most relevant proteolytic systems that together with chaperones contribute to creating the proteostasis network that is kept in dynamic balance to maintain overall functionality of cellular proteomes. Data accumulated over decades demonstrate that the effectiveness of elements of the proteostasis network declines with age. In this scenario, failure to degrade misfolded or faulty proteins increases the risk of protein aggregation, chronic inflammation, and the development of age-related diseases. This is especially important in the context of aging-related modification of functions of the immune system

Homogenization and enhancement for the G-equation

We consider the so-called G-equation, a level set Hamilton-Jacobi equation, used as a sharp interface model for flame propagation, perturbed by an oscillatory advection in a spatio-temporal periodic environment. Assuming that the advection has suitably small spatial divergence, we prove that, as the size of the oscillations diminishes, the solutions homogenize (average out) and converge to the solution of an effective anisotropic first-order (spatio-temporal homogeneous) level set equation. Moreover we obtain a rate of convergence and show that, under certain conditions, the averaging enhances the velocity of the underlying front. We also prove that, at scale one, the level sets of the solutions of the oscillatory problem converge, at long times, to the Wulff shape associated with the effective Hamiltonian. Finally we also consider advection depending on position at the integral scale

Relaxation Methods for Mixed-Integer Optimal Control of Partial Differential Equations

We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynamics of the system by choosing, for each instant in time, one of the actuators together with ordinary controls. We consider relaxation techniques that are already used successfully for mixed-integer optimal control of ordinary differential equations. Our analysis yields sufficient conditions such that the optimal value and the optimal state of the relaxed problem can be approximated with arbitrary precision by a control satisfying the integer restrictions. The results are obtained by semigroup theory methods. The approach is constructive and gives rise to a numerical method. We supplement the analysis with numerical experiments