Applied Mathematics and Optimization最新文献

The Keller–Segel system is a classical model in chemotaxis, widely used in biological and physical contexts, but also a challenging prototype for nonlinear PDE analysis. Our focus studies an insensitizing control problem for the nonlinear parabolic-parabolic Keller–Segel system, which models chemotactic behavior in biological systems. Our goal is to find a control that makes a certain functional of the solution insensitive to small perturbations in the initial data. We show that this problem is equivalent to achieving partial null controllability for a related cascade system that reflects the main structure of the original dynamics. Thanks to this equivalence, we focus on analyzing the controllability of the cascade system. We begin by studying the linearized version of the problem. Using a duality approach, along with carefully selected weighted estimates and energy techniques, we establish a suitable observability inequality. This key result enables us to move on to the nonlinear case. We address the nonlinear system through a local inverse mapping argument, relying on the continuity and differentiability of the control-to-state map in an appropriate functional framework, along with other key assumptions.

This paper deals with the following system with signal-dependent motility

under homogeneous Neumann boundary conditions in a smooth bounded domain (Omega subset mathbb {R}^n) ((nge 2)). Here, (a,b > 0), (l>2,gamma >0) and (frac{l}{gamma }>frac{n+2}{2}), the function (phi in C^2([0,infty ))) satisfies (phi (s)>0) for all (sge 0), and (varphi (s)=(alpha - 1)phi '(s)) with (alpha in (0,1)), then the considered system possesses a global classical solutions which are uniformly bounded.

In this article, we establish Carleman estimates for degenerate wave operators, focusing on cases where degeneracy occurs only on a portion of the boundary. Under specific geometrical assumptions, we construct novel weight functions tailored to address the challenges posed by the degeneracy, thereby deducing global Carleman estimates. As applications, we prove the exact controllability (both boundary and internal) for the degenerate wave equation using the Hilbert Uniqueness Method.

Entry-exit dynamics are crucial in modeling crowd movement. Here, we present a novel first-order, stationary mean-field game (MFG) model on bounded domains that accurately captures entry-exit dynamics. In our model, the interior dynamics are governed by a standard first-order stationary MFG system: a first-order Hamilton-Jacobi equation coupled with a transport equation. The model incorporates mixed boundary conditions that correspond to an entry region (Gamma _N) and an exit region (Gamma _D). A Neumann condition on (Gamma _N) prescribes the agent inflow via a non-homogeneous flux term, (j(x)); a no-entry condition on (Gamma _D) restricts this boundary region to exit only, preventing inward flow; finally, in (Gamma _D), we prescribe an upper bound on the exit cost combined with a complementary contact-set condition. This contact-set condition identifies boundary points where the value function attains the exit cost (contact points) versus points where the non-penetration condition prevents artificial inflows (non-contact points). However, as our examples show, contact does not necessarily imply that exit occurs. This mixed approach overcomes the limitations of classical Dirichlet conditions, which can artificially force boundary points to act as both entry and exit sites. We analyze the system using a variational formulation, applying the direct method of calculus of variations to establish the existence of solutions under minimal regularity assumptions. Furthermore, we prove the uniqueness of the gradient of the value function (particularly in regions with positive agent density) and the uniqueness of the density function. Several examples, including cases in one and two dimensions, illustrate first-order MFG phenomena such as the formation of empty regions (where agent density vanishes) and the proper assignment of entry and exit roles. These results establish a rigorous mathematical foundation for modeling realistic entry-exit scenarios.

In this paper, building on the formulation of quantum Markov decision processes (q-MDPs) presented in our previous work [N. Saldi, S. Sanjari, and S. Yüksel, Quantum Markov Decision Processes: General Theory, Approximations, and Classes of Policies, SIAM Journal on Control and Optimization, 2024], our focus shifts to the development of semi-definite programming approaches for optimal policies and value functions of both open-loop and classical-state-preserving closed-loop policies. First, by using the duality between the dynamic programming and the semi-definite programming formulations of any q-MDP with open-loop policies, we establish that the optimal value function is linear and there exists a stationary optimal policy among open-loop policies. Then, using these results, we establish a method for computing an approximately optimal value function and formulate computation of optimal stationary open-loop policy as a bi-linear program. Next, we turn our attention to classical-state-preserving closed-loop policies. Dynamic programming and semi-definite programming formulations for classical-state-preserving closed-loop policies are established, where duality of these two formulations similarly enables us to prove that the optimal policy is linear and there exists an optimal stationary classical-state-preserving closed-loop policy. Then, similar to the open-loop case, we establish a method for computing the optimal value function and pose computation of optimal stationary classical-state-preserving closed-loop policies as a bi-linear program.

The objective of this article is to consider the initial-value problem derived from an extended quasilinear May-Nowak system in a two-dimensional smoothly bounded domain, encompassing viral kinetics with particular emphasis on scenarios captured by two dominant mechanisms: the cross-diffusive behavior of healthy individuals toward the orientation of the infected populations, and the nonlinear diffusion process in the form (nabla cdot (D(r)nabla r)) among fluid environment. Here, the diffusion term D(r) represents a slight generalization of the prototypical expression ((1+r)^{m-1}) with (rge 0). Under the optimal assumption when (m>1) in parabolic-parabolic-elliptic framework, then for arbitrary choice of the initial datum, the corresponding problem possesses at least one globally bounded weak solution. Insofar as we are aware, this is the first result to reveal the complex interplay between cross-diffusion dynamic, nonlinear diffusion process and fluid coupling mechanism of such system, therefore effectively improving the regularity of solutions without compromising global well-posedness.

Existence of the random attractors for a nonlocal weak damping wave equation driven by nonlinear colored noise is investigated in a bounded domain, where the nonlinear terms f(u) and h(t, x, u) in the equation are critical growth. First, the global well-posedness of solutions is established using the theory of monotone operators. Second, the existence of a random absorbing set is proved via energy estimates. Meanwhile, we extend the method of contraction functions verifying the pullback asymptotic compactness of non-autonomous hyperbolic systems from the deterministic case to the random case. With the aid of above theoretical findings, we further obtain the pullback asymptotic compactness of the random dynamical system associated with the problem. Ultimately, existence of the random attractors is shown. It’s worth mentioning that the abstract conclusions of [39] are extended from the deterministic systems to the random ones. Moreover, we employ the weaker nonlinearity conditions in this paper than in [19]. In order to deal with the critical growth of nonlinear function and nonlinear colored noise, we seek out a useful Bihari-type integral inequality introduced in [16], which helps us overcome the difficulty caused by the critical growth of two nonlinear terms.

We propose a comprehensive framework for solving constrained variational inequalities via various classes of evolution equations displaying multi-scale aspects. In an infinite-dimensional Hilbertian framework, the class of dynamical systems we propose combine Tikhonov regularization and exterior penalization terms in order to induce strong convergence of trajectories to least norm solutions in the constrained domain. Our construction thus unifies the literature on regularization methods and penalty-based dynamical systems. An extension to a full splitting formulation of the constrained domain is also provided, with associated weak convergence results involving the Attouch-Czarnecki condition.

It is shown that a switching control involving a finite number of Dirac delta actuators is able to steer the state of a general class of nonautonomous parabolic equations to zero as time increases to infinity. The strategy is based on a recent feedback stabilizability result, which utilizes control forces given by linear combinations of appropriately located Dirac delta distribution actuators. Then, the existence of a stabilizing switching control with no more than one actuator is active at each time instant is established. For the implementation in practice, the stabilization problem is formulated as an infinite-horizon optimal control problem, with cardinality-type control constraints enforcing the switching property. Subsequently, this problem is tackled using a receding horizon framework. Its suboptimality and stabilizing properties are analyzed. Numerical simulations validate the approach, illustrating its stabilizing and switching properties.

We investigate a double phase problem on non-compact manifolds by leveraging orbit expansions of isometry groups. A revised Ambrosetti-Rabinowitz condition is proposed, and we prove that the problem admits a nontrivial solution and infinitely many solutions, respectively.