We study the Stackelberg-Nash null controllability of a coupled system governed by two linear forward stochastic parabolic equations. The system includes one leader control localized in a subset of the domain, two additional leader controls in the diffusion terms, and m follower controls, where m & ecaron; 2. We consider two different scenarios for the followers: first, when the followers minimize a functional involving both components of the system's state, and second, when they minimize a functional involving only the second component of the state. For fixed leader controls, we first establish the existence and uniqueness of the Nash equilibrium in both scenarios and provide its characterization. As a byproduct, the problem is reformulated as a classical null controllability issue for the associated coupled forward-backward stochastic parabolic system. To address this, we derive new Carleman estimates for the adjoint stochastic systems. As far as we know, this problem is among the first to be discussed for stochastic coupled systems.

This paper presents the concepts of exact, null, and approximate controllability in the Stackelberg-Nash sense for abstract forward and backward stochastic evolution equations, involving two types of controls: leaders and followers. We begin by proving the existence and uniqueness of the Nash equilibrium, as well as its characterization for fixed leader controls. We then establish a duality between these controllability concepts and the corresponding observability properties. Finally, we apply our theoretical results to the forward and backward stochastic heat equations. The results for the backward heat equation are obtained by deriving a new Carleman estimate.

We investigate the robust Stackelberg null controllability of a one-dimensional forward linear stochastic Kuramoto–Sivashinsky–Korteweg–de Vries (KS–KdV) equation. The control framework is formulated as a hierarchical Stackelberg game involving two leaders, one follower, and worst-case disturbances acting in both the drift and diffusion terms. The first leader acts to drive the system to rest, while the second leader is introduced to overcome analytical difficulties arising from the stochastic setting. The follower, by reducing the effect of the disturbances, addresses a tracking-type control problem aimed at keeping the system state and its first and second spatial derivatives close to prescribed target trajectories. First, the robust control problem is characterized by the existence of a saddle point. Then, the analysis is reduced to the null controllability of a strongly coupled forward–backward stochastic KS–KdV system. The problem is addressed by combining a duality technique with new Carleman estimates for forward and backward stochastic fourth-order parabolic equations.

In this paper, we establish the existence of insensitizing controls for forward linear stochastic parabolic equations with dynamic boundary conditions. We begin by reducing the insensitizing control problem to a classical controllability property of a cascade system of coupled forward-backward stochastic parabolic equations. Next, we derive a suitable observability inequality for the adjoint system using a new Carleman estimate. Finally, employing the classical duality arguments, we solve the considered control problem.

We study a hierarchical control problem for stochastic parabolic equations that involve a gradient term in the drift part. We employ the Stackelberg-Nash strategy with two leaders and two followers. The leaders are responsible for selecting the policy targeting null controllability, while the followers solve a bi-objective optimal control problem which consists of maintaining the solution process close to prefixed targets. Once the Nash equilibrium is determined, the problem reduces to achieving null controllability of a coupled forward-backward stochastic system. To solve this problem, via Carleman estimates, we establish a suitable observability inequality. Subsequently, we achieve the desired controllability result.

In this paper, we study a multi-objective inverse initial problem with a Nash strategy constraint for forward stochastic reaction-diffusion equations with dynamic boundary conditions, where both the volume and surface equations are influenced by randomness. The objective is twofold: first, we maintain the state close to prescribed targets in fixed regions using two controls; second, we determine the history of the solution from observations at the final time. To achieve this, we establish new Carleman estimates for forward and backward equations, which are used to prove an interpolation inequality for a coupled forward-backward stochastic system. Consequently, we obtain two results: backward uniqueness and a conditional stability estimate for the initial conditions.

This paper deals with a hierarchical multi-objective control problem for forward stochastic parabolic equations with dynamic boundary conditions. The controls are divided into two classes: leaders and followers. The goal of the leaders is of null controllability type while the followers are in charge of letting the state close to prescribed targets in fixed observation regions. To solve the problem, Nash and Stackelberg strategies are used. To implement these strategies, we combine some appropriate Carleman estimates and the well-known control duality approach.

This paper deals with a hierarchical control problem for the heat equation with dynamic boundary conditions. The main goal consists of letting the state near from a prescribed target in a fixed observation region. The secondary objective is the null controllability. In other words, we reverse the roles of the leader and the follower addressed in the recent article. For this purpose, we combine some appropriate Carleman estimates and the Stackelberg strategy. We also extend our study for hierarchical-biobjective problems by applying the Stackelberg-Pareto strategy.

The paper deals with time and norm optimal control problems for the heat equation with dynamic boundary conditions in the context of the null controllability. More precisely, we answer an open question left in our previous paper (Boutaayamou et al., in Math Methods Appl Sci 45:1359–1376, 2021). To do so, we first prove a new Lebeau–Robbiano spectral inequality using a logarithmic convexity inequality, then an observability inequality on any set of positive measure. This plays a relevant role in proving the existence and uniqueness of optimal null controls. Finally, the connection between time and norm optimal null controls is presented.

In this paper, we deal with time and norm optimal control problems for the heat equation with dynamic boundary conditions. We prove the existence of admissible controls and, combining a suitable Carleman estimate for such an equation with some analyticity results, we show a strong unique continuation property. Then, with the aid of this unique continuation, bang‐bang properties and the connection between time and norm optimal controls can be established.

This paper deals with the null controllability of the semilinear heat equation with dynamic boundary conditions of surface diffusion type, with nonlinearities involving drift terms. First, we prove a negative result for some function \begin{document}$ F $\end{document} that behaves at infinity like \begin{document}$ |s| \ln ^{p}(1+|s|), $\end{document} with \begin{document}$ p > 2 $\end{document}. Then, by a careful analysis of the linearized system and a fixed point method, a null controllability result is proved for nonlinearties \begin{document}$ F(s, \xi) $\end{document} and \begin{document}$ G(s, \xi) $\end{document} growing slower than \begin{document}$ |s| \ln ^{3 / 2}(1+|s|+\|\xi\|)+\|\xi\| \ln^{1 / 2}(1+|s|+\|\xi\|) $\end{document} at infinity.

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.

In this paper, we study an inverse problem for linear parabolic system with variable diffusion coefficients subject to dynamic boundary conditions. We prove a global Lipschitz stability for the inverse problem involving a simultaneous recovery of two source terms from a single measurement and interior observations, based on a recent Carleman estimate for such problems.

We consider the heat equation with dynamic bounary conditions involving gradient terms in a bounded domain. In this paper we study the cost of approximate controllability for this equation. Combining new developed Carleman estimates and some optimization techniques, we obtain explicit bounds of the minimal norm control. We consider the linear and the semilinear cases.