Uniform stability of semilinear wave equations with arbitrary local memory effects versus frictional dampings

This paper is concerned with the mixed initial–boundary value problem for semilinear wave equations with complementary frictional dampings and memory effects. We successfully establish uniform exponential and polynomial decay rates for the solutions to this initial–boundary value problem under much weak conditions concerning memory effects. More specifically, we obtain the exponential and polynomial decay rates after removing the fundamental condition that the memory-effect region includes a part of the system boundary, while the condition is a necessity in the previous literature; moreover, for the polynomial decay rates we only assume minimal conditions on the memory kernel function g, without the usual assumption of $g^{\prime }$ controlled by g.     

Free boundary problem for a fully nonlinear and degenerate parabolic equation in an angular domain

This paper concerns with the problem of how to running an insurance company to maximize his total discounted expected dividends. In our model, the dividend rate is limited in $[0,M]$ and the company is allowed to transfer any proportion of risk by reinsuring. So there are two strategies which we call dividend strategy and reinsurance strategy. The objective function and the corresponding optimal two strategies are the solution and the two free boundaries of the following Barenblatt parabolic equation

$v_t?\underset{0\le a\le 1}{\max }?(\frac{1}{2}{a}^2{\sigma }^2{v}_{xx}+a\mu v_x)+cv?\underset{0\le l\le M}{\max }?[(1?v_x)l]=0$

under certain boundary conditions on an angular domain

$Q_T=\{(x,t)|0<x<Mt,0<t\le T\}.$

The main effort is to analyze the properties of the solution and the free boundaries to show the optimal decision for the insurance company.     

Controllability results for the Moore–Gibson–Thompson equation arising in nonlinear acoustics

We show that the Moore–Gibson–Thomson equation

$\tau {?}_{ttt}y+\alpha {?}_{tt}y?c^2\mathrm{\Delta }y?b\mathrm{\Delta }{?}_{t}y=k{?}_{tt}(y^2)+{\chi }_{\omega (t)}u,$

is controlled by a force that is supported on an moving subset $\omega (t)$ of the domain, satisfying a geometrical condition. Using the concept of approximately outer invertible map, a generalized implicit function theorem and assuming that $\gamma :=\alpha ?\frac{\tau c^2}{b}>0$, the local null controllability in the nonlinear case is established. Moreover, the analysis of the critical value $\gamma =0$ for the linear equation is included.     

A central limit theorem in nonlinear filtering

It is shown that the probability law of a diffusion process conditioned on weakly corrupted observations is asymptotically Gaussian when properly scaled. The method of proof involves Fisher information matrices and a Cramér-Rao inequality.     

A variational approach to second-order approximability of sliding mode control systems

We consider robustness properties of second-order methods for the sliding mode control of nonlinear ordinary differential equations. A new approach is presented based on the theory of well-posed optimization problems. It is shown that the convergence of the real states of the control system to the ideal one is intimately related to Tykhonov well-posedness of suitably defined dynamic optimization problems.     

Normal form for linear systems with respect to its vector relative degree

For multi-input multi-output (MIMO) linear systems with existing vector relative degree a normal form is constructed. This normal form is not only structural simple but allows to characterize the system’s zero dynamics for the design of feedback controllers. A characterization of the zero dynamics in terms of the normal form is given.     

Optimal proportional reinsurance and investment based on Hamilton-Jacobi-Bellman equation

In the whole paper, the claim process is assumed to follow a Brownian motion with drift and the insurer is allowed to invest in a risk-free asset and a risky asset. In addition, the insurer can purchase the proportional reinsurance to reduce the risk. The paper concerns the optimal problem of maximizing the utility of terminal wealth. By solving the corresponding Hamilton-Jacobi-Bellman equations, the optimal strategies about how to purchase the proportional reinsurance and how to invest in the risk-free asset and risky asset are derived respectively.     

A PDE approach to regularity of solutions to finite horizon optimal switching problems

We study optimal 2-switching and n-switching problems and the corresponding system of variational inequalities. We obtain results on the existence of viscosity solutions for the 2-switching problem for various setups when the cost of switching is non-deterministic. For the n-switching problem we obtain regularity results for the solutions of the variational inequalities. The solutions are C^1,1-regular away for the free boundaries of the action sets.     

Weak maximum principle and accessory problem for control problems on time scales

In this paper we derive the first and second variations for a nonlinear time scale optimal control problem with control and state-endpoints equality constraints. Using the first variation, a first order necessary condition for weak local optimality is obtained under the form of a weak maximum principle generalizing the Dubois–Reymond Lemma to the optimal control setting and time scales. A second order necessary condition in terms of the accessory problem is derived by using the nonnegativity of the second variation at all admissible directions. The control problem is studied under a controllability assumption, and with or without the shift in the state variable. These two forms of the problem are shown to be equivalent.     

Optimality conditions with state constraints for semilinear systems determined by operator valued measures

In this paper we develop the necessary conditions of optimality for a class of distributed parameter systems (partial differential equations) determined by operator valued measures and controlled by vector measures. Based on some recent results on existence of optimal controls from the space of vector measures, we develop necessary conditions of optimality for a class of control problems. The main results are the necessary conditions of optimality for problems without state constraints and those with state constraints. Also, a conceptual algorithm along with a brief discussion of its convergence is presented.