Stackelberg Nash Strategies Research Articles

Hierarchical null controllability of a semilinear degenerate parabolic equation with a gradient term

ABSTRACT In this paper, we apply a hierarchical strategy to a semilinear weakly degenerate parabolic equation with non-linearity that depends on the solution of the system and the spatial derivative of the solution. We use the Stackelberg–Nash strategy with one leader aiming to drive the solution to zero and two followers intended to solve a Nash equilibrium corresponding to a bi-objective optimal control problem. Since the system is semilinear, the functionals are not convex in general. To overcome this difficulty, we first prove the existence and uniqueness of the Nash quasi-equilibrium, which is a weaker formulation of the Nash equilibrium. Next, with additional conditions, we establish the equivalence between the concepts of Nash quasi-equilibrium and Nash equilibrium. We establish a suitable Carleman inequality for the adjoint system and then an observability inequality. Based on this observability inequality, we prove the null controllability of the linearized system. Then, using Kakutani's fixed point Theorem, we demonstrate the null controllability of the main system.

Hierarchical exact controllability of a parabolic equation with boundary controls

This paper addresses the application of a multi-objective control problem to a parabolic equation. We present the Stackelberg-Nash strategy, which combines the concepts of controllability to trajectories with Nash equilibrium. Our assumption is that the system is influenced through a hierarchy of boundary controls, consisting of one main control (the leader) aiming to drive the solution to a specified target at a final time, and a pair of secondary controls (the followers) designed to minimize two prescribed cost functionals while adapting to the leader's objective. The main novelty of this work lies in the fact that all controls are located on the boundary. One of the primary challenges encountered is the derivation of suitable observability inequality of Carleman for an adjoint system with boundary coupling terms.

Hierarchical control for the semilinear parabolic equations with interior degeneracy

This paper concerns with the hierarchical control of the semilinear parabolic equations with interior degeneracy. By a Stackelberg-Nash strategy, we consider the linear and semilinear system with one leader and two followers. First, for any given leader, we analyze a Nash equilibrium corresponding to a bi-objective optimal control problem. The existence and uniqueness of the Nash equilibrium is proved, and its characterization is given. Then, we find a leader satisfying the null controllability problem. The key is to establish a new Carleman estimate for a coupled degenerate parabolic system with interior degeneracy.

Hierarchical Control for the Oldroyd Equation in Memoriam to Professor Luiz Adauto Medeiros.

This manuscript deals with a hierarchical control problem for Oldroyd equation under the Stackelberg-Nash strategy. The Oldroyd equation model is defined by non-regular coefficients, that is, they are bounded measurable functions. We assume that we can act in the dynamic of the system by a hierarchy of controls, where one main control (the leader) and several additional secondary control (the followers) act in order to accomplish their given tasks: controllability for the leader and optimization for followers. We obtain the existence and uniqueness of Nash equilibrium and its characterization, the approximate controllability with respect to the leader control, and the optimality system for leader control.

Null controllability and Stackelberg–Nash strategy for a $2\times 2$ system of parabolic equations

This paper is dedicated to solving a multi-objective control problem for a 2\times2 system of parabolic equations. Here, we have many objectives, possibly conflictive, and we adopt a concept of hierarchy for the controls. We have the leader control, responsible for objectives of controllability type, and the follower controls which intend to be a Nash-equilibrium with respect to some given cost functionals. The novelty here is that we formulate this problem for systems of parabolic equations, meaning that we have many variables and naturally much more objectives to accomplish.

Hierarchical exact controllability of hyperbolic equations and Dunford-Pettis' theorem

This paper deals with the Stackelberg-Nash strategies for boundary control problems for linear and semilinear wave equations. Assuming that we can act on the system through a hierarchy of controls, to each leader, we associate a Nash equilibrium (two followers) corresponding to a bi-objective optimal control problem. Then we look for a leader that solves an exact boundary controllability problem. We mainly consider the case of a semilinear hyperbolic PDE where all the controls (leader and followers) act on small parts of the boundary. In view of the lack of regularity, the arguments usually invoked to prove controllability do not work here. Accordingly, we carry out a fixed-point approach based on weak compactness in $L^1$. To this purpose, we apply Dunford-Pettis' Theorem. We also consider the case where the followers act in small subsets of the domain.

Stackelberg–Nash Null Controllability for a Non Linear Coupled Degenerate Parabolic Equations

The main purpose of this paper is to apply the notion of hierarchical control to a coupled degenerate non linear parabolic equations. We use the Stackelberg–Nash strategy with one leader and two followers. The followers solve a Nash equilibrium corresponding to a bi-objective optimal control problem and the leader a null controllability problem. Since the considered problem is non linear, the associated cost is non-convex. We first prove the existence, uniqueness and the characterization of the Nash quasi-equilibrium, which is a weak formulation of the Nash equilibrium because the cost associated to the non linear problem is non-convex. Next, we show that under suitable conditions, the Nash quasi-equilibrium is equivalent to the Nash equilibrium. Finally using some Carleman inequalities that we established, and the Kakutani’s fixed point Theorem, we brough the states of our system to the rest at final time T.

Stackelberg-Nash exact controllability for the Kuramoto-Sivashinsky equation with boundary and distributed controls

This paper deals with a multi-objective control problem for the Kuramoto-Sivashinsky equation by following a Stackelberg-Nash strategy. We have a distributed control called Leader, and two boundary controls called Followers, each of them has to act over the equation to influence the behavior of the state in a particular way, by reaching or approaching to many targets at once. To be more precise, the Leader wants to drive the solution to a prescribed target at a final time, and the followers have to minimize some given cost functionals, adapting themselves to what the Leader wants. The main difficulty here is that, since the Followers are in the boundary, the problem turns to be equivalent to prove a partial null controllability result for a system of nonlinear fourth-order equations with boundary coupling terms.

Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions

<p style='text-indent:20px;'>This paper deals with the hierarchical control of the anisotropic heat equation with dynamic boundary conditions and drift terms. We use the Stackelberg-Nash strategy with one leader and two followers. To each fixed leader, we find a Nash equilibrium corresponding to a bi-objective optimal control problem for the followers. Then, by some new Carleman estimates, we prove a null controllability result.</p>

Hierarchic control for a nonlinear parabolic equation in an unbounded domain

In this paper, we study the hierarchical control using the Stackelberg–Nash strategy for a nonlinear parabolic equation in an unbounded domain. We assume that we can act on the system by three controls hierarchically. Two controls called followers that provide a Nash equilibrium for two cost functional. The third control named leader is supposed to bring the state of the system to rest at the final time. The results are achieved by means of observability inequality of Carleman type that we established for the adjoint systems and a fixed point theorem under the assumption that the uncontrolled domain is bounded.

Stackelberg-Nash Controllability for a Quasi-linear Parabolic Equation in Dimension 1D, 2D, or 3D

This paper deals with the application of Stackelberg-Nash strategies to the control to quasi-linear parabolic equations in dimensions 1D, 2D, or 3D. We consider two followers, intended to solve a Nash multi-objective equilibrium; and one leader satisfying the controllability to the trajectories.

Controllability for a One-Dimensional Wave Equation in a Non-cylindrical Domain

This paper deals with the controllability for a one-dimensional wave equation with mixed boundary conditions in a non-cylindrical domain. This equation models small vibrations of a string where an endpoint is fixed and the other is moving. As usual, we consider one main control (the leader) and an additional secondary control (the follower). We use Stackelberg–Nash strategies.

Hierarchical exact controllability of semilinear parabolic equations with distributed and boundary controls

We present some exact controllability results for parabolic equations in the context of hierarchic control using Stackelberg–Nash strategies. We analyze two cases: in the first one, the main control (the leader) acts in the interior of the domain and the secondary controls (the followers) act on small parts of the boundary; in the second one, we consider a leader acting on the boundary while the followers are of the distributed kind. In both cases, for each leader, an associated Nash equilibrium pair is found; then, we obtain a leader that leads the system exactly to a prescribed (but arbitrary) trajectory. We consider linear and semilinear problems.

Stackelberg–Nash exact controllability for the Kuramoto–Sivashinsky equation

This paper deals with a hierarchical control problem for the Kuramoto–Sivashinsky equation following a Stackelberg–Nash strategy. We assume that there is a main control, called the leader, and two secondary controls, called the followers. The leader tries to drive the solution to a prescribed target and the followers intend to be a Nash equilibrium for given functionals. It is known that this problem is equivalent to a null controllability result for an optimality system consisting of three non-linear equations. One of the novelties is a new Carleman estimate for a fourth-order equation with right-hand sides in Sobolev spaces of negative order, which allows to relax some geometric conditions for the observation sets for the followers.

Stackelberg–Nash null controllability for some linear and semilinear degenerate parabolic equations

This paper deals with the application of Stackelberg–Nash strategies to the null controllability of degenerate parabolic equations. We assume that we can act on the system through a hierarchy of controls. A first control (the leader) is assumed to determine the policy; then, a Nash equilibrium pair (corresponding to a noncooperative multi-objective optimization strategy) is found; this governs the action of other controls (the followers). This way, the state of the system is driven to zero and, consequently, we solve a hierarchical null controllability problem. The main novelty in this paper is that the physical systems are governed by linear or semilinear 1D heat equations with degenerate coefficients.

Hierarchic Control for the Wave Equation

This paper deals with the hierarchical control of the wave equation. We use Stackelberg–Nash strategies. As usual, we consider one leader and two followers. To each leader we associate a Nash equilibrium corresponding to a bi-objective optimal control problem; then, we look for a leader that solves an exact controllability problem. We consider linear and semilinear equations.

New results on the Stackelberg–Nash exact control of linear parabolic equations

This paper is concerned with Stackelberg–Nashstrategies to control parabolic equations. We have one control, the leader, that is responsible for a null controllability property; additionally, we have a couple of controls, called the followers, that provides a Nash equilibrium for two cost functionals. This is a classical situation in many fields of science and, in mathematics, leads to a lot of interesting questions and open problems and possesses many applications. In the main result, we prove the existence of a leader such that the corresponding controlled system is driven to zero. This way, we improve some questions that were left open in previous works.

Hierarchic control for a coupled parabolic system

In this paper we study a Stackelberg–Nash strategy to control systems of coupled linear parabolic partial differential equations. We assume that we can act on the system through several controls called followers , intended to solve a Nash multi-objective equilibrium, and a single leader control satisfying a null controllability objective. We obtain the existence and uniqueness of the Nash equilibrium, its characterization and the optimal leader control satisfying the null controllability problem.

Stackelberg–Nash exact controllability for linear and semilinear parabolic equations

This paper deals with the application of Stackelberg–Nash strategies to the control of parabolic equations. We assume that we can act on the system through a hierarchy of controls. A first control (the leader) is assumed to choose the policy. Then, a Nash equilibrium pair (corresponding to a noncooperative multiple-objective optimization strategy) is found; this governs the action of the other controls (the followers). The main novelty in this paper is that, this way, we can obtain the exact controllability to a prescribed (but arbitrary) trajectory. We study linear and semilinear problems and, also, problems with pointwise constraints on the followers.

On the approximate controllability of Stackelberg–Nash strategies for linear heat equations in ℝN with potentials

In this paper, we establish hierarchic control for the linear heat equation in ℝN with potentials. Our strategy is inspired by the techniques developed by Díaz and Lions [On the approximate controllability of Stackelberg-Nash strategies. In: Díaz JI, editor. Ocean circulation and pollution control mathematical and numerical investigations. Berlin: Springer; 2005. p. 17–27]; however, many new difficulties arise due to lack of compactness of Sobolev embeddings. We manage these adversities by employing similarity variables and weighted Sobolev spaces.