This paper is concerned with a linear quadratic optimal control for a class of singular Volterra integral equations. Under proper convexity conditions, optimal control uniquely exists, and it could be characterized via Frechet derivative of the quadratic functional in a Hilbert space or via maximum principle type necessary conditions. However, these (equivalent) characterizations have a shortcoming that the current value of the optimal control depends on the future values of the optimal state. Practically, this is not feasible. The main purpose of this paper is to obtain a causal state feedback representation of the optimal control.
A linear-quadratic optimal control problem for a forward stochastic Volterra integral equation (FSVIE, for short) is considered. Under the usual convexity conditions, open-loop optimal control exists, which can be characterized by the optimality system, a coupled system of an FSVIE and a Type-II backward SVIE (BSVIE, for short). To obtain a causal state feedback representation for the open-loop optimal control, a path-dependent Riccati equation for an operator-valued function is introduced, via which the optimality system can be decoupled. In the process of decoupling, a Type-III BSVIE is intr…
This paper is concerned with a class of controlled singular Volterra integral equations, which could be used to describe problems involving memories. The well-known fractional order ordinary differential equations of the Riemann--Liouville or Caputo types are strictly special cases of the equations studied in this paper. Well-posedness and some regularity results in proper spaces are established for such kind of questions. For the associated optimal control problem, by using a Liapounoff's type theorem and the spike variation technique, we establish a Pontryagin's type maximum principle for op…
In this paper, we study the optimal control problem with terminal and inequality state constraints for state equations described by Volterra integral equations having singular and nonsingular kernels. The singular kernel introduces abnormal behavior of the state trajectory with respect to the parameter of $α\in (0,1)$. Our state equation is able to cover various state dynamics such as any types of Volterra integral equations with nonsingular kernels only, fractional differential equations (in the sense of Riemann-Liouville or Caputo), and ordinary differential state equations. We obtain the we…
This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations, where the solution X^{u,ξ}(t)=X(t) is given by X(t) =ϕ(t)+\int_{0}^{t}}b(t,s,X(s),u(s)) ds+\int_{0}^{t}σ(t,s,X(s),u(s))dB(s) +\int _{0}^{t}\int_{0}^{t}h(t,s) dξ(s). Here dB(s) denotes the Brownian motion Itô type differential and ξdenotes the singular control (singular in time t with respect to Lebesgue measure) and u denotes the regular control (absolutely continuous with respect to Lebesgue measure). Such systems may for example be used to model harvesting of populations with memo…
We uncover the gradient structure to investigate the convergence of solutions in nonlocal nonlinear dynamical systems. Mainly but not exclusively, we use the Lojasiewicz inequality to prove convergence results in various spaces with continuous, or discrete temporal domain, and finite, or infinite dimensional spatial domain. To be more specific, we analyze Lotka-Volterra type dynamics and concentration-dispersion dynamics. Lotka-Volterra equations describe the population dynamics of a group of species. Under the assumption that the interaction between species is symmetric, we present two differ…
