Dan Goreac

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Verified

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• PhD, HDR
• Professor at Université Laval

We aim at characterizing domains of attraction for controlled piecewise deterministic processes using an occupational measure formulation and Zubov's approach. Firstly, we provide linear programming (primal and dual) formulations of discounted, infinite horizon control problems for PDMPs. These formulations involve an infinite-dimensional set of pr...

We aim at studying the property of controlled stochastic flows with mean-field dynamics to comply with some (closed) state restrictions. This property, known as (near)-viability, is tackled via (quasi-)tangency methods. Law restrictions and mixed state-law restrictions are considered as the interplay between the two classes. As an auxiliary result...

This paper focuses on the characterization of viability zones in compartmental models with varying population size, due both to deaths caused by epidemics and to natural demography. This is achieved with the use of viscosity characterizations of viability and extensively illustrated on several models. An example taking into consideration real data...

This paper is intended as a companion to [12] and focuses on the numerical implementations of the method in the context of insurance problems with mixed controls: reinsurance acting as continuous control, and capital injection and dividend payment acting as singular ones. The aim is twofold. On the one hand, we provide a comparison of the output of...

This paper’s aim is threefold. First, using Feynman’s path approach to the derivation of the classical Schrödinger’s equation in Feynman (Rev Mod Phys 20:367-387, 1948) and by introducing a slight path (or wave) dependency of the action, we derive a new class of equations of Schrödinger type where the driving operator is no longer the Laplace one b...

In this paper we explore several novel notions of exact controllability for mean-field linear controlled stochastic differential equations (SDEs). A key feature of our study is that the noise coefficient is not required to be of full rank. We begin by demonstrating that classical exact controllability with $\mathbb{L}^2$-controls necessarily requir...

The aim of the present paper is to investigate the optimal vaccination policies with prevalence restrictions in an SIRS demographic model. We provide a well-posedness result for the system and give a thorough description of safety zones (immunity and feasible) when intensive care units (ICU) restrictions are enforced on the prevalence. Using Pontry...

This paper studies an optimal control problem for a class of SIR epidemic models, in scenarios in which the infected population is constrained to be lower than a critical threshold imposed by the intensive care unit (ICU) capacity. The vaccination effort possibly imposed by the health-care deciders is classically modeled by a control input affectin...

The aim of the present paper is to investigate the optimal vaccination policies with prevalence restrictions in an SIRS demographic model. We provide a well-posedness result for the system and give a thorough description of safety zones (immunity and feasible) when intensive care units (ICU) restrictions are enforced on the prevalence. Using Pontry...

We consider optimal control problems which consist in minimizing the $L^\infty$ norm of an output function under an isoperimetric or $L^{1}$ inequality. These problems typically arise in control applications when one looks to minimizing the maximum trajectory deviation or “peak” under a budget constraint. We show a duality with more classical p...

This paper studies an optimal control problem for a class of SIR epidemic models, in scenarios in which the infected population is constrained to be lower than a critical threshold imposed by the ICU (intensive care unit) capacity. The vaccination effort possibly imposed by the health-care deciders is classically modeled by a control input affectin...

We aim at studying a novel mathematical model associated to a physical phenomenon of infiltration in an homogeneous porous medium. The particularities of our system are connected to the presence of a gravitational acceleration term proportional to the level of saturation, and of a Brownian multiplicative perturbation. Furthermore, the boundary cond...

We prove that the solution to the singular-degenerate stochastic fast-diffusion equation with parameter m ∈ (0,1), with zero Dirichlet boundary conditions on a bounded domain in any spatial dimension, and driven by linear multiplicative Wiener noise, exhibits improved regularity in the Sobolev space W^{1,m+1}_0 for initial data in L^2.

A very large class of ODE epidemic models (2) discussed in this paper enjoys the property of admitting also an integral renewal formulation, with respect to an "age of infection kernel" a(t) which has a matrix exponential form (6). We observe first that a very short proof of this fact is available when there is only one susceptible compartment , an...

This paper studies a two-scales Markov compartmental model (SIRS) with demography, vaccination, reinfections and waning immunity, and with external control inputs of pharmaceutical (vaccination) and non-pharmaceutical (social distancing) type. The demography events are considered to be rare and contribute to a jump mechanism with trajectory dependi...

A very large class of ODE epidemic models (2.2) discussed in this paper enjoys the property of admitting also an integral renewal formulation, with respect to an "age of infection kernel" a(t) which has a matrix exponential form (3.2). We observe first that a very short proof of this fact is available when there is only one susceptible compartment,...

The aim of this paper is to provide a rigorous mathematical analysis of an optimal control problem of a SIR epidemic on an infinite horizon. A state constraint related to intensive care units (ICU) capacity is imposed and the objective functional linearly depends on the state and the control. After preliminary asymptotic and viability analyses, a $...

We provide a duality result linking the value function for a control problem with supremum cost H under an isoperimetric inequality G $\le$ gmax, and the value function for the same controlled dynamics with cost G and state constraint H $\le$ hmax. This duality is proven for initial conditions at which lower semi-continuity of the value functions c...

This paper is concerned with the existence and uniqueness of the solution for the stochastic fast logarithmic equation with Stratonovich multiplicative noise in $\mathbb{R}^{d}$ for $d\geqslant 3$. It provides an answer to a critical case (morally speaking, corresponding to the porous media operator $\Delta X^m$ for $m=0$) left as an open problem i...

We aim at providing a characterization of the ability to maintain a stochastic coupled system with porous media components in a prescribed set of constraints by using internal controls. This property is proven via a quasi-tangency local-in-time condition in the spirit of Euler approximation schemes. In particular, by employing one of the components...

In this work, we first introduce a class of deterministic epidemic models with varying populations inspired by Arino et al. (2007), the parameterization of two matrices, demography, the waning of immunity, and vaccination parameters. Similar models have been focused on by Julien Arino, Fred Brauer, Odo Diekmann, and their coauthors, but mostly in t...

The aim of the present paper is to provide necessary and sufficient conditions to maintain a stochastic coupled system with porous media components and gradient‐type noise in a prescribed set of constraints by using internal controls. This work is a complementary contribution to the results obtained by the same authors, also on the viability proble...

The aim of this paper is twofold. On one hand, we strive to give a simpler proof of the optimality of greedy controls when the cost of interventions is control-affine and the dynamics follow a state-constrained controlled SIR model. This is achieved using the Hamilton–Jacobi characterization of the value function, via the verification argument and...

This paper focuses on linearisation techniques for a class of mixed singular/continuous control problems and ensuing algorithms. The motivation comes from (re)insurance problems with reserve-dependent premiums with Cram{\'e}r-Lundberg claims, by allowing singular dividend payments and capital injections. Using variational techniques and embedding t...

This paper focuses on linearisation techniques for a class of mixed singular/continuous control problems and ensuing algorithms. The motivation comes from (re)insurance problems with reserve-dependent premiums with Cramér-Lundberg claims, by allowing singular dividend payments and capital injections. Using variational techniques and embedding the t...

The aim of this paper is to provide a rigorous mathematical analysis of an optimal control problem with SIR dynamics. The main feature of our study is the presence of state constraints (related to intensive care units ICU capacity) and strict target objectives (related to the immunity threshold). The first class of results provides a comprehensive...

The recent papers Gajek and Kucinsky (Insur Math Econ 73:1–19, 2017) and Avram et al. (Mathematics 9(9):931, 2021) cost induced dichotomy for optimal dividends in the cramr-lundberg model. Avram et al. (Mathematics 9(9):931, 2021) investigated the control problem of optimizing dividends when limiting capital injections stopped upon bankruptcy. The...

The aim of the present paper is to provide necessary and sufficient conditions to maintain a stochastic coupled system, with porous media components and gradient-type noise in a prescribed set of constraints by using internal controls. This work is a continuation of the results in [10], as we consider the case of divergence type noise perturbation....

We aim at providing a characterization of the ability to maintain a stochastic coupled system with porous media components in a prescribed set of constraints by using internal controls. This property is proven via a quasi-tangency local-in-time condition in the spirit of Euler approximation schemes. In particular, by employing one of the components...

In this work we study the stability properties of the equilibrium points of deterministic epidemic models with nonconstant population size. Models with nonconstant population have been studied in the past only in particular cases, two of which we review and combine. Our main result shows that for simple "matrix epidemic models" introduced in [1], a...

This paper is motivated by the asymptotic stabilization of abstract SPDEs of linear type. As a first step, it proposes an abstract contribution to the exact controllability (in a general $\mathbb{L}^p$-sense, $p>1$) of a class of linear SDEs with general time-invariant rank control coefficient in the diffusion term. From this point of view, our pap...

The aim of this paper is to provide a rigorous mathematical analysis of an optimal control problem with SIR dynamics. The main feature of our study is the presence of state constraints (related to intensive care units ICU capacity) and strict target objectives (related to the immunity threshold). The first class of results provides a comprehensive...

We investigate a control problem leading to the optimal payment of dividends in a Cramér-Lundberg-type insurance model in which capital injections incur proportional cost, and may be used or not, the latter resulting in bankruptcy. For general claims, we provide verification results, using the absolute continuity of super-solutions of a convenient...

We study an optimal control problem consisting in minimizing the L ∞ norm of a Borel measurable cost function, in finite time, and over all trajectories associated with a controlled dynamics which is reflected in a compact prox-regular set. The first part of the paper provides the viscosity characterization of the value function for uniformly conti...

In the context of maximizing cumulative dividends under barrier policies, generalized Azéma–Yor (draw-down) stopping times receive increasing attention during these past years. Based on Pontryagin’s maximality principle, we illustrate the necessity of such generalizations under the framework of spectrally negative Markov processes. Roughly speaking...

The recent papers Gajek-Kucinsky(2017) and Avram-Goreac-Li-Wu(2020) investigated the control problem of optimizing dividends when limiting capital injections stopped upon bankruptcy. The first paper works under the spectrally negative L\'evy model; the second works under the Cram\'er-Lundberg model with exponential jumps, where the results are cons...

Motivated by the result of invariance of regular-boundary open sets in [P. Cannarsa, G. D. Prato, and H. Frankowska, Indiana Univ. Math. J., 59 (2010), pp. 53--78] and multi-stability issues in gene networks, our paper focuses on three closely related aims. First, we give a necessary local Lipschitz-like condition in order to expect invariance of o...

In this paper, we study a stochastic control problem faced by an insurance company allowed to pay out dividends and make capital injections. As in (Løkka and Zervos (2008); Lindensjö and Lindskog (2019)), for a Brownian motion risk process, and in Zhu and Yang (2016), for diffusion processes, we will show that the so-called Løkka–Zervos alternative...

This paper deals with some reachability issues for piecewise linear switched systems with time-dependent coefficients and multiplicative noise. Namely, it aims at characterizing data that are almost reachable at some fixed time \(T>0\) (belong to the closure of the reachable set in a suitable \({\mathbb {L}}^2\)-sense). From a mathematical point of...

The first motivation of our paper is to explore further the idea that, in risk control problems, it may be profitable to base decisions both on the position of the underlying process Xt and on its supremum X¯t:=sup0≤s≤tXs. Strongly connected to Azema-Yor/generalized draw-down/trailing stop time this framework provides a natural unification of draw-...

The first motivation of our paper is to explore further the idea that, in risk control problems, it may be profitable to base decisions both on the position of the underlying process Xt and on its supremum Xt := sup 0$\le$s$\le$t Xs. Strongly connected to Azema-Yor/generalized draw-down/trailing stop time (see [AY79]), this framework provides a nat...

This paper deals with some reachability issues for piecewise linear switched systems with time-dependent coefficients and multiplicative noise. Namely, it aims at characterizing data that are almost reachable at some fixed time T > 0 (belong to the closure of the reachable set in a suitable L 2-sense). From a mathematical point of view, this provid...

Motivated by the result of invariance of regular-boundary open sets in \cite{CannarsaDaPratoFrankowska2009} and multi-stability issues in gene networks, our paper focuses on three closely related aims. First, we give a necessary local Lipschitz-like condition in order to expect invariance of open sets (for deterministic systems). Comments on optima...

We investigate some stability properties of approximate null-controllability (A0C). We show that A0C is inherited when increasing the number of observed jumps. However, in all generality, there is no direct connection between A0C with an initial mode and the A0C starting from directly accessible modes. We have also given a sufficient condition unde...

We consider stochastic differential systems driven by a Brownian motion and a Poisson point measure where the intensity measure of jumps depends on the solution. This behavior is natural for several physical models (such as Boltzmann equation, piecewise deterministic Markov processes, etc). First, we give sufficient conditions guaranteeing that the...

We consider stochastic differential systems driven by a Brownian motion and a Poisson point measure where the intensity measure of jumps depends on the solution. This behavior is natural for several physical models (such as Boltzmann equation, piecewise deterministic Markov processes, etc). First, we give sufficient conditions guaranteeing that the...

This paper aims at the study of controllability properties and induced controllability metrics on complex networks governed by a class of (discrete time) linear decision processes with mul-tiplicative noise. The dynamics are given by a couple consisting of a Markov trend and a linear decision process for which both the '' deterministic '' and the n...

We investigate a mathematical model associated to the infection time in multistable gene networks. The mathematical processes are of hybrid switch type. The switch is governed by pure jump modes and linked to DNA bindings. The differential component follows backward stochastic dynamics reflected in some mode-dependent nonconvex domains. First, we s...

In this paper we study the exact null-controllability property for a class of controlled PDMP of switch type with switch-dependent, piecewise linear dynamics and multiplicative jumps. First, we show that exact null-controllability induces a con-trollability metric. This metric is linked to a class of backward stochastic Riccati equations. Using arg...

We study optimal control problems in infinite horizon when the dynamics belong to a specific class of piecewise deterministic Markov processes constrained to star-shaped networks (inspired by traffic models). We adapt the results in [H. M. Soner. Optimal control with state-space constraint. II. SIAM J. Control Optim., 24(6):1110.1122, 1986] to prov...

We propose an explicit, easily-computable algebraic criterion for approximate null-controllability of a class of general piecewise linear switch systems with multiplicative noise. This gives an answer to the general problem left open in [13]. The proof relies on recent results in [4] allowing to reduce the dual stochastic backward system to a famil...

We propose an explicit, easily-computable algebraic criterion for approximate null-controllability of a class of general piecewise linear switch systems with multiplicative noise. This gives an answer to the general problem left open in [13]. The proof relies on recent results in [4] allowing to reduce the dual stochastic backward system to a famil...

In this note we show that, in the context of stochastic control systems, the uniform existence of a limit of Cesàro averages implies the existence of uniform limits for averages with respect to a wide class of measures dominated by the Lebesgue measure and satisfying some asymptotic condition. It gives a partial answer to the problem mentioned in R...

We aim at studying approximate null-controllability properties of a particular class of piecewise linear Markov processes (Markovian switch systems). The criteria are given in terms of algebraic invariance and are easily computable. We propose several necessary conditions and a sufficient one. The hierarchy between these conditions is studied via s...

This paper aims at the study of controllability properties and induced controllability metrics on complex networks governed by a class of (discrete time) linear decision processes with mul-tiplicative noise. The dynamics are given by a couple consisting of a Markov trend and a linear decision process for which both the "deterministic" and the noise...

We aim at characterizing the asymptotic behavior of value functions in the control of piece-wise deterministic Markov processes (PDMP) of switch type under nonexpansive assumptions. For a particular class of processes inspired by temperate viruses, we show that uniform limits of discounted problems as the discount decreases to zero and time-average...

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 fi...

In this short paper we prove that, in the framework of continuous control problems for piecewise deterministic Markov processes, the existence of a uniform limit for discounted value functions as the discount factor vanishes implies (without any further assumption) the uniform convergence of the value functions with long run average cost as the tim...

We prove a uniform Abelian result for controlled systems with piecewise deterministic Markov dynamics : the existence of a uniform limit for value functions with discounted costs as the discount factor decreases to zero implies the existence of a (uniform) value function with long time average cost. The result is independent of dissipativity proper...

We propose a linearized formulation for min–max control problems with separated dynamics. First, we investigate the existence of the value function and saddle points for semicontinuous costs. Second, we obtain dual formulations and dynamic programming principles. Copyright © 2013 John Wiley & Sons, Ltd.

We study some controllability properties for linear stochastic systems of mean-field type. First, we give necessary and sufficient criteria for exact terminal-controllability. Second, we characterize the approximate and approximate null-controllability via duality techniques. Using Riccati equations associated to linear quadratic problems in the co...

We consider an optimal stochastic control problem for which the payoff is the average of a given cost function. In a non ergodic setting, but under a suitable nonexpansivity condition, we obtain the existence of the limit value when the averaging parameter converges (namely the discount factor tends to zero for Abel mean or the horizon tends to inf...

The aim of this synthesis is to present my research activity covering the period elapsed since the terminal year of my PhD program (i.e. the period October 2008 - February 2013). My subjects of research roughly correspond to three main directions, each one presented in a separate section : ∙ Linear programming (LP) methods in deterministic and stoc...

This paper aims at studying a class of discontinuous deterministic control problems under state constraints using a linear programming approach. As for classical control problems (Gaitsgory and Quincampoix (2009) [16], Goreac and Serea (2011) [19]), the primal linear problem is stated on some appropriate space of probability measures. Naturally, th...

We study a class of nonlinear stochastic control problems with semicontinuous cost and state constraints using a linear programming (LP) approach. First, we provide a primal linearized problem stated on an appropriate space of probability measures with support contained in the set of constraints. This space is completely characterized by the coeffi...

We present linearization techniques for the Mayer stochastic control problem and the L∞-control problem. These techniques proved to be very useful in dealing with deterministic and stochastic control problems.

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 se...

We investigate the approximate controllability property for a class of linear stochastic equations driven by independent Brownian motion and Poisson random measure. The paper generalizes recent results of Buckdahn et al. (2006, A characterization of approximately controllable linear stochastic differential equations. Stochastic Partial Differential...

The aim of the paper is to provide a linearization approach to the L 1 -control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the Dynamic Programming Principle. Then, we use the L p approach a...

Using the linear programming approach to stochastic control introduced by Buckdahn, Goreac, and Quincampoix, and by Goreac and Serea, we provide a semigroup property for some set of probability measures leading to dynamic programming principles for stochastic control problems. An abstract principle is provided for general bounded costs. Linearized...

In this paper, we study a criterion for the viability of stochastic semi-linear control systems on a real, separable Hilbert space. The necessary and sufficient conditions are given using the notion of stochastic quasi-tangency. As a consequence, we prove that approximate viability and the viability property coincide for stochastic linear control s...

We study two classes of stochastic control problems with semicontinuous cost: the Mayer problem and optimal stopping for controlled diffusions. The value functions are introduced via linear optimization problems on appropriate sets of probability measures. These sets of constraints are described deterministically with respect to the coefficient fun...

We study a classical stochastic optimal control problem with constraints and discounted payoff in an infinite horizon setting. The main result of the present paper lies in the fact that this optimal control problem is shown to have the same value as a linear optimization problem stated on some appropriate space of probability measures. This enables...

The aim of this paper is to study two classes of discontinuous control problems without any convexity assumption on the dynamics. In the first part we characterize the value function for the Mayer problem and the supremum cost problem using viscosity tools and the notion of ε-viability (near viability). These value functions are given with respect...

We aim at characterizing viability, invariance and some reachability properties of controlled piecewise deterministic Markov processes (PDMPs). Using analytical methods from the theory of viscosity solutions, we establish criteria for viability and invariance in terms of the first order normal cone. We also investigate reachability of arbitrary ope...

In this paper we study a criterion for the viability of stochastic semilinear control systems on a real, separable Hilbert space. The necessary and sufficient conditions are given using the notion of stochastic quasi-tangency. As a consequence, we prove that approximate viability and the viability property coincide for stochastic linear control sys...

The objective of the paper is to investigate the approximate controllability property of a linear stochastic control system with values in a separable real Hilbert space. In a first step we prove the existence and uniqueness for the solution of the dual linear backward stochastic differential equation. This equation has the particularity that in ad...

The aim of this paper is to introduce an insurance model allowing reinsurance and dividend payment. Our model deals with several homogeneous contracts and takes into account the legislation regarding the provisions to be justified by the insurance companies. This translates into some restriction on the (maximal) number of contracts the company is a...

We are interested in the approximate controllability property for a linear stochastic differential equation. For deterministic control necessary and sufficient criterion exists and is called Kalman condition. In the stochastic framework criteria are already known either when the control fully acts on the noise coefficient or when there is no contro...

The aim of this thesis is to present some contributions in the framework of control for stochastic differential equations in finite or infinite dimension: (1) Non-compact-valued stochastic control under state constraints; The first chapter is devoted to the study of a necessary condition under which the solutions of a stochastic differential equati...

In the present paper, we study a necessary condition under which the solutions of a stochastic differential equation governed by unbounded control processes, remain in an arbitrarily small neighborhood of a given set of constraints. We prove that, in comparison to the classical constrained control problem with bounded control processes, a further a...

We study approximate controllability for a linear stochastic differential equation dy(t)=(Ay(t)+Bu(t))dt+(Cy(t)+Du(t))dW(t), for the case when the control acts also on the noise. This may be considered as a generalization of the work of Buckdahn, Quincampoix and Tessitore where the problem is solved for D=0 and of Peng for D of full rank. We prove,...

Le but de cette thèse est de présenter quelques contributions dans le cadre du contrôle des équations différentielles stochastiques en dimension finie où infinie : (1) Contrôle stochastique non borné sous contraintes d'état. Nous étudions une condition nécessaire sous laquelle les solutions d'une EDS régie par un processus de contrôle non-borné res...