We study an infinite horizon stochastic optimal control problem by means of the associated Hamilton-Jacobi-Bellman equation. The problem we are studying has the particularity of having a discount factor which can take both signs, depending on the value of the state.
We study an optimal control problem over a finite time horizon, where an individual makes decisions on consumption, investment, and retirement timing in continuous time. After choosing to retire early, the individual must satisfy a minimum consumption level. The agent determines consumption and portfolio allocations jointly with the retirement decision, aiming to achieve the highest possible expected utility over their lifetime. To address this problem, we employ the martingale duality approach to characterize the optimal strategies and determine the free boundary that delineates the retirement and working regions. This boundary is characterized by a nonlinear integral equation, which we solve numerically using a recursive scheme. Our computational method provides accurate evaluations of the optimal strategies. The findings emphasize how subsistence consumption requirements influence the timing of retirement and investment behavior.
. In this work, we established a unique continuation result and the well-posedness in Sobolev-type spaces of negative order for a special onedimensional system that models the evolution of long water waves with small amplitude. The property of unique continuation was proved using a Carlemantype estimate, and the argument for the well-posedness combined estimates of the Strichartz type and the Fourier transform restriction method.
. In this paper, we study the existence of the 0(h)-insensitizing controls for semi-discrete stochastic parabolic equations of both second- and fourth-order. Here, the concept of semi-discrete refers to the spatial variable discretization via a finite difference scheme with mesh size h, while the time variable remains continuous, and the 0(h)-insensitizing control problem is a weaker notion of the insensitization problem depending on the step-size h. The 0(h)-insensitizing control problem for semi-discrete stochastic parabolic equations (second- and fourth-order, respectively) is reduced to the 0-null controllability property for the cascade systems of forward-backward semi-discrete stochastic parabolic equations. We first establish the new semi-discrete Carleman estimates for the dual of the cascade systems, and further show the relaxed observability inequalities of these dual systems. Moreover, the 0-null controllability property for the cascade systems of forward-backward semi-discrete stochastic parabolic equations is obtained via the classical duality arguments.
. In this research, we propose a novel control scheme to compensate for the effects of arbitrarily long input delays in heterodirectional hyperbolic partial differential equation systems with zero transport speed. Based on the delay phenomenon in the transport equation, the input delay is first transformed into a new transport equation, resulting in an equivalent system without delay. The controller is then designed using the backstepping method, in which the backstepping transformation consists of two classical second-type Volterra transformations and one affine-Volterra transformation. Unlike the Volterra transformation kernels, which are defined on triangular domains, the affine-Volterra transformation kernel is defined on a square domain. Proving the well-posedness of this kernel is the main challenge encountered in this work. Moreover, the presence of zero speed renders the invertibility of the affine-Volterra transformation less straightforward. With additional efforts, we demonstrate its invertibility. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed control scheme.
. This paper is concerned with a class of second-order evolution ory, and a constant time delay feedback within a real Hilbert space. While many previous studies have addressed stability in autonomous settings, this study focuses on the nonautonomous case of delayed systems. The main results presented in the paper are the well-posedness and stabilization of solutions under a globally Lipschitz continuous nonlinear source term and a constant delay. Under suitable assumptions on time-dependent operators, we prove the system is well-posed by semigroup approach. The stabilization is established following the construction of an appropriate Lyapunov functional and the application of the energy method. The result is new for such time-delayed nonautonomous systems with infinite memory and nonlinear effects.
An optimal harvesting control problem for the McKendrick-von Foerster equation with generic cost functional is considered. The notions of the normal cone and the tangent cone are used to establish the necessary optimality conditions. The existence and uniqueness of an optimal control are proved with the aid of the Ekeland variational principle. Moreover, a synthesis of the optimal feedback law is given.
. In this paper, we analyze a distributed optimal control problem associated to the strong solutions of a 3D Boussinesq system with Navier-slip boundary conditions. Since the existence of strong (global) solutions is an open question, we consider a suitable cost functional such that any weak solution, satisfying a regularity criterion, is also a global-in-time strong solution. Then, we prove the existence of optimal solutions and deduce the differentiability of the map control-to-state via the implicit function theorem. As a consequence, we obtain first-order necessary optimality conditions for local optimal solutions.
In this paper, the initial boundary value problem of the Kortewegde Vries Burgers equation on the negative half-plane is analyzed. Initially, the well-posedness on H-s(R-) for s > -1 of the IBVP is established to concentrate on the L-2(R-) controllability problem when the controls are in the Dirichlet and Newmann conditions at x = 0.
This study primarily focuses on establishing the sufficient conditions for the existence and uniqueness of the mild solution along with the approximate and trajectory controllability results for a new class of the non-linear Psi-Caputo fractional neutral-type integro-differential system with finite delay and nonlocal conditions in a Hilbert space. A key advantage of the Psi-Caputo fractional derivative is that it allows to choose a suitable kernel function Psi. First, we derive the existence of the mild solution for the proposed control system by using a fixed point approach. For this purpose, the proposed control system is transformed into an equivalent fixed point problem using the Psi-Riemann-Liouville fractional integral operator. Then, the existence of the mild solution is established by Schauder's fixed point theorem. Then, the uniqueness of the mild solution is studied with the help of the Banach contraction principle. Moreover, the approximate controllability result of the proposed control system is established under the consideration that the corresponding linear system is approximate controllable. Further, the trajectory controllability result is studied by using the Gronwall's inequality. The set of sufficient conditions is derived by using the concepts of fractional calculus, Laplace transform, fixed point techniques, and semigroup theory of bounded linear operators. Finally, an illustrative example is presented to validate and demonstrate the applicability of the theoretical results.
This paper addresses a discrete-time optimal control problem under the worst-case scenario for the controller, where its control policies are subject to specific restrictions at defined costs. The system dynamics follows a hybrid evolution, incorporating both regular and impulsive sub-dynamics that drive the state through time. The goal is to minimize a total cost criterion in the worst-case scenario, with costs discounted over time by a state-actiondependent discount factor that can reach the values of zero or one at certain points. The problem is approached through an auxiliary family of unrestricted minimax control problems, each indexed by a particular parameter. For each member of this family, we determine the optimal solution, and then seek the "best" parameter that aligns the unrestricted problem with the original one. This alignment ensures that the optimal solutions for the unrestricted problems also apply to the restricted version. To illustrate our theory, we provide an example based on a transboundary pollution accumulation problem.
This paper is concerned with stochastic impulse control problems in which the running cost changes depending on the impulse control. Because of such a dependence, it brings several difficulties when the usual dynamic programming principle is used. The corresponding Hamilton-Jacobi-Bellman (HJB) equation (a quasi-variational inequality) was derived, which contains a parameter. The value function is a unique viscosity solution to this HJB equation by a classical argument. Further, inspired by the derivation of the Pontryagin-type maximum principle for stochastic optimal controls with a nonconvex control domain, we have established the maximum principle for our stochastic optimal impulse controls, allowing perturbations in optimal impulse moments.
. This paper develops a stochastic maximum principle for general control systems driven by mixed Brownian motions (mBms) (including the standard Brownian motion and the fractional Brownian motion with Hurst parameter in (1/2, 1) ). The control domain need not be convex. We establish rigorous estimates for both first-order and second-order variational equations, which form the foundation for deriving necessary conditions of the maximum principle. By combining the martingale representation theorem with properties of transformation operators specific to fractional Brownian motion, we construct a novel family of backward stochastic differential equations (BSDEs) that characterize optimal controls. As an application, we solve a linear-quadratic (LQ) control problem under this general framework, demonstrating both the theoretical validity and the practical applicability of the main results.
This paper aims to explore the robust reinsurance contract design with belief heterogeneity in the framework of the Stackelberg differential game. In this setup, the reinsurer plays the role of leader to decide the optimal reinsurance premium, while the insurer acts as the follower to determine the optimal retention level under proportional reinsurance arrangements. It is assumed that the insurer and the reinsurer have heterogeneous beliefs and adopt distinct distributions for the aggregate risk due to asymmetric information and different estimation methods. Moreover, we suppose that the reinsurer applies interval estimation to approximate the expected value of aggregate claims, denoted as mu 2, within an interval [mu 2 - delta, mu 2 + delta], where mu 2 represents the estimated value and delta signifies the estimation error. Under the time-consistent mean-variance criterion, we derive reinsurance contracts for both ambiguityaverse and ambiguity-loving reinsurers by solving the corresponding extended Hamilton-Jacobi-Bellman systems. Furthermore, to provide a deeper understanding of the influence of heterogeneous beliefs on the optimal reinsurance contract and value functions, we focus on delta = 0 and derive several intuitive results.
. We consider optimal control problems for the two-dimensional stationary Navier-Stokes equation with cost functionals involving the pressure, stress, diffusion, and convection. Likewise, we study observations for the velocity, pressure, and stress, concentrated on a finite collection of points located either in the domain or on the boundary. Such observations are known to produce PDEs with measure data for the corresponding adjoint equation. Depending on the nature of the objective cost functional, the control set will be either the space of square-integrable functions without constraints, with constraints in Lebesgue spaces, or with weak derivatives. We prove generalized Green's theorems for the weak and very weak solutions that involve the trace and normal stress on the boundary. Finally, the local optimality systems for such control problems will be derived and the regularity of the optimal states will be established.
In this paper, we use discrete stochastic feedback control to stabilize an unstable system. We prove that there exists a discrete step size tau(-) > 0 with tau < tau(-) such that the responding stochastically controlled system is mean square and quasi-sure exponentially stable. We give an example to show the validity of the control strategy.
We study a stochastic optimal control problem with the state constrained to a smooth, compact domain. The control influences both the drift and a possibly degenerate, control-dependent dispersion matrix, leading to a fully nonlinear, degenerate elliptic Hamilton–Jacobi–Bellman (HJB) equation with a nontrivial Neumann boundary condition. Although these features have been studied separately, this work provides the first unified treatment combining them all. We establish that the optimal value function associated with the control problem is the unique viscosity solution of the HJB equation with a nontrivial Neumann boundary condition, and we present an illustrative example demonstrating the applicability of the framework.
We study the Pontryagin maximum principle by deriving necessary and sufficient conditions for a class of optimal control problems arising in non exchangeable mean field systems, where agents interact through heterogeneous and asymmetric couplings. Our analysis leads to a collection of forward-backward stochastic differential equations (FBSDE) of non exchangeable mean field type. Under suitable assumptions, we establish the solvability of this system. As an illustration, we consider the linear-quadratic case, where the optimal control is characterized by an infinite dimensional system of Riccati equations.
In this work, we investigate the approximate controllability of a class of one-dimensional degenerate parabolic equations with Robin boundary conditions. The degeneracy occurs at one endpoint of the spatial domain, and we apply an impulsive control in a small region at a fixed moment. Our main result establishes an observability inequality for the adjoint system, from which we deduce approximate controllability at final time . The proof relies on a logarithmic convexity argument, developed through a Carleman commutator approach.
