Applied Mathematics and Optimization最新文献

This paper presents a two-player stochastic differential game with diffusion control and rewards that are zero-sum. The players have i.i.d. exponentially distributed time horizons at which their states are compared and only the best player receives a reward. We assume that players have no information about their opponent and can only use private information for controlling their state. We show that there always exists a Markovian saddle point. Moreover, we consider the existence of saddle points in the class of threshold controls, i.e., controls choosing the maximal volatility below some threshold and the minimal above. There exists a symmetric saddle point of threshold type in closed form if and only if the ratio of maximal and minimal volatility does not exceed the value (sqrt{2}).

Motivated by an asset-liability management problem and an optimal investment of N agents under relative performance criteria with common noise, this paper studies optimal control problems of conditional mean field type under partial and full observation. For the partial observation case, we remove the restriction that observation coefficient is uniformly bounded and allow it to grow linearly (unbounded) with respect to state variable, which gives rise to some difficulties in the subsequent analysis. Different with the conventional approximate method (indirect method), we derive a maximum principle with linear observation coefficient adopting a dominated growth rate idea (direct method). As adjoint equation, a conditional mean field backward stochastic differential equation with stochastic Lipschitz coefficients is introduced. Combining backward separation method with state-augmentation technique, a closed form candidate optimal premium strategy of asset-liability management problem is derived. For the full observation case, by virtue of the derived maximum principle and mean field game theory, we investigate a conditional mean-variance portfolio selection problem and obtain the efficient frontier and efficient portfolio explicitly, which is proved to be an (epsilon )-Nash equilibrium of the optimal investment of N agents under relative performance concern with common noise. Some numerical simulations with sound financial interpretations are presented, which verify the effectiveness of our theoretical results.

In this paper, we consider a varying terminal time structure for the stochastic optimal control problem under state constraints, in which the terminal time varies with the mean value of the state. In this new stochastic optimal control system, the control domain does not need to be convex and the diffusion coefficient contains the control variable. To overcome the difficulty in the proof of the related Pontryagin’s stochastic maximum principle, we develop asymptotic first- and second-order adjoint equations for the varying terminal time, and establish its variational equation. In the end, we present two examples to verify the main results of this study.

In this paper, we study infinite horizon overtaking optimal control problems for fractional system involving linear quadratic performance index which is allowed to be unbounded. When the controllability or stabilizability condition is not assumed and the nonhomogeneous term and the weight function are global integrability, it is shown that the solution of the overtaking optimal control can be derived by solving the linear quadratic optimal control problem with the state equation and running cost rate function in the controllable subspace based on the orthogonal projection. When the nonhomogeneous term and the weight function are not global integrability, both the existence and nonexistence of solutions for overtaking optimal control in various cases are established by characterizing the weight function. Several examples are also included to verify the above-mentioned theory. There results reveal that the overtaking optimality approach can be adopted to solve some optimal control problems with unbounded performance index and at the same time, the limitation of this approach is also exposed.

This study extends previous work (Özer in 63rd IEEE conference on decision and control, pp 8498–8503, 2024) on the stabilization of piezoelectric beam systems with a dynamic tip mass, emphasizing collocated partial damping designs. Unlike earlier approaches based on noncollocated controllers and Lyapunov methods, this work investigates the performance of collocated controllers under reduced damping configurations. A distinction is made between higher order damping, associated with the strain rate, and lower order damping, associated with the tip velocity and the accumulated electrical current at the electrodes. It is shown that higher order damping provides stronger stabilization compared to tip velocity feedback. Conditions for exponential and polynomial decay rates are identified in relation to the damping configuration and material properties. A rigorous analytical framework is introduced to characterize these decay behaviors without relying on Lyapunov techniques or spectral analysis. Numerical simulations support the theoretical findings, highlighting the essential role of the higher-order feedback mechanism in achieving exponential stability. The results offer practical insights for applications in energy harvesting, acoustic wave control, and high-precision sensing.

This paper studies a discounted linear-quadratic (LQ) leader-follower stochastic differential game for regime switching diffusion in an infinite horizon. Within the (L^{2,r})-stabilizability framework, we first, as a preliminary, establish the global well-posedness of infinite horizon linear stochastic differential equations and backward stochastic differential equations with Markov chains. Next, under the uniform convexity condition for LQ problems, we obtain an open-loop Stackelberg equilibrium of the leader-follower game. By employing the so-called four-step scheme, the corresponding Hamiltonian systems for the two players are decoupled and then the open-loop Stackelberg equilibrium admits a state feedback representation in terms of two new-type algebraic Riccati equations together with some certain stabilizing condition. Finally, we report a numerical example to illustrate our theoretical results, including the solutions to the Riccati equations, the Stackelberg equilibrium strategies, and the behavior of the corresponding state process.

We prove the existence of a global martingale solution of a stochastic electroconvection equations on ({mathbb {R}}^2) with multiplicative noise. The proof is based on the stochastic compactness method and the Jakubowski generalization of the Skorokhod theorem. The main difficulty is caused by the nonlinearity term qRq which makes the equations strongly nonlinear.

In this paper, we study a two-species chemotaxis–Navier–Stokes system with Lotka–Volterra type competitive kinetics: n_t + u · ∇n = Δn − χ₁∇ · (n∇w) + n(λ₁ − μ₁n^θ−1 − a₁v); v_t + u · ∇v = Δv − χ₂∇ · (v∇w) + v(λ₂ − μ₂v − a₂n); w_t + u · ∇w = Δw − w + n + v; u_t + κ(u · ∇)u = Δu + ΔP + (n + v)∇ϕ; (nabla cdot u=0), (xin Omega ), (t>0) in a bounded and smooth domain (Omega subset {mathbb {R}}^2) with no-flux/Dirichlet boundary conditions, where (chi _1, chi _2) are positive constants. We present the global existence of generalized solution to a two-species chemotaxis–Navier–Stokes system and the eventual smoothness already occurs in systems with much weaker degradation ((theta >1)), again under a smallness condition on (lambda _1, lambda _2).

Stabilization of linear control systems with parameter-dependent system matrices is investigated. A Riccati based feedback mechanism is proposed and analyzed. It is constructed by means of an ensemble of parameters from a training set. This single feedback stabilizes all systems of the training set and also systems in its vicinity. Moreover its suboptimality with respect to optimal feedback for each single parameter from the training set can be quantified.

In this paper, we consider an optimal control problem of an ordinary differential inclusion governed by the hypergraph Laplacian, which is defined as a subdifferential of a convex function and then is a set-valued operator. We can assure the existence of optimal control for a suitable cost function by using methods of a priori estimates established in the previous studies. However, due to the multivaluedness of the hypergraph Laplacian, it seems to be difficult to derive the necessary optimality condition for this problem. To cope with this difficulty, we introduce an approximation operator based on the approximation method of the hypergraph, so-called “clique expansion.” We first consider the optimality condition of the approximation problem with the clique expansion of the hypergraph Laplacian and next discuss the convergence to the original problem. In appendix, we state some basic properties of the clique expansion of the hypergraph Laplacian for future works.