Oana-Silvia Serea

On Reflecting Boundary Problem for Optimal Control

This paper deals with Mayers problem for controlled systems with reflection on the boundary of a closed subset K. The main result is the characterization of the possibly discontinuous value function in terms of a unique solution in a suitable sense to a partial differential equation of Hamilton--Jacobi--Bellman type.

Computation of viability kernels: a case study of by-catch fisheries

Traditional means of studying environmental economics and management problems consist of optimal control and dynamic game models that are solved for optimal or equilibrium strategies. Notwithstanding the possibility of multiple equilibria, the models’ users—managers or planners—will usually be provided with a single optimal or equilibrium strategy no matter how reliable, or unreliable, the underlying models and their parameters are. In this paper we follow an alternative approach to policy making that is based on viability theory. It establishes “satisficing” (in the sense of Simon), or viable, policies that keep the dynamic system in a constraint set and are, generically, multiple and amenable to each manager’s own prioritisation. Moreover, they can depend on fewer parameters than the optimal or equilibrium strategies and hence be more robust. For the determination of these (viable) policies, computation of “viability kernels” is crucial. We introduce a MATLAB application, under the name of VIKAASA, which allows us to compute approximations to viability kernels. We discuss two algorithms implemented in VIKAASA. One approximates the viability kernel by the locus of state space positions for which solutions to an auxiliary cost-minimising optimal control problem can be found. The lack of any solution implies the infinite value function and indicates an evolution which leaves the constraint set in finite time, therefore defining the point from which the evolution originates as belonging to the kernel’s complement. The other algorithm accepts a point as viable if the system’s dynamics can be stabilised from this point. We comment on the pros and cons of each algorithm. We apply viability theory and the VIKAASA software to a problem of by-catch fisheries exploited by one or two fleets and provide rules concerning the proportion of fish biomass and the fishing effort that a sustainable fishery’s exploitation should follow.

Differential Games and Zubov's Method

In this paper we provide generalizations of Zubovs equation to differential games without the Isaacs condition. We show that both generalizations of Zubovs equation (which we call the min-max and max-min Zubov equation, respectively) possess unique viscosity solutions which characterize the respective controllability domains. As a consequence, we show that under the usual Isaacs condition the respective controllability domains as well as the local controllability assumptions coincide.

Optimality Issues for a Class of Controlled Singularly Perturbed Stochastic Systems

The present paper aims at studying stochastic singularly perturbed control systems. We begin by recalling the linear (primal and dual) formulations for classical control problems. In this framework, we give necessary and sufficient support criteria for optimality of the measures intervening in these formulations. Motivated by these remarks, in a first step, we provide linearized formulations associated with the value function in the averaged dynamics setting. Second, these formulations are used to infer criteria allowing to identify the optimal trajectory of the averaged stochastic system.

Reflected Differential Games

This paper concerns the existence of a possible discontinuous value for a zero sum two-player reflected differential game under Isaacs condition. We characterize the value function as the unique solution—in a suitable sense—to a variational inequality, namely, the Hamilton-Jacobi-Isaacs differential inclusion.

Some Applications of Linear Programming Formulations in Stochastic Control

We present two applications of the linearization techniques in stochastic optimal control. In the first part, we show how the assumption of stability under concatenation for control processes can be dropped in the study of asymptotic stability domains. Generalizing Zubov’s method, the stability domain is then characterized as some level set of a semicontinuous generalized viscosity solution of the associated Hamilton–Jacobi–Bellman equation. In the second part, we extend our study to unbounded coefficients and apply the method to obtain a linear formulation for control problems whenever the state equation is a stochastic variational inequality.

Some support considerations in the asymptotic optimality of two-scale controlled PDMP

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Some Support Considerations in the Asymptotic Optimality of Two-Scale Controlled PDMP D Goreac, Oana Silvia Serea