Narrow Results

In this paper, we consider a nonlocal backward heat equation in a bounded domain Ω ⊂ ℝⁿ which has a Lipschitz boundary. The well-posedness is well known for the nonlocal forward heat equation, but the corresponding backward equation is ill-posed, at least in ordinary spaces. In this paper, we obtain the well-posedness of the nonlocal backward heat equation in a dense subspace of L²(Ω). Based on the well-posedness result, we obtain a singular optimal system (SOS) for the nonlocal backward heat system.

Suppose that one party proposes to another a contract for sharing an uncertain profit which maximizes the former’s expected utility, with respect to its beliefs, subject to a constraint on the latter’s expected utility, with respect to the latter’s beliefs. It turns out that the optimal contract, which we find, can be nonmonotone, as well as nonlinear, in the realized profit. To avoid the implausible lack of monotonicity, we formulate and solve a model constrained to have monotone increasing profits for both partners. If beliefs are identical, the (unconstrained) contract is shown to be monotone, and under certain conditions, linear. That might explain one famous contract from the history of jazz. If the other party can be assumed risk neutral, the linear contract reduces to the former receiving a constant amount, and the latter the residual net profit, as in the case of another famous contract from the history of jazz. Since in the type of partnerships, we have in mind the partners are always motivated to exert high effort due to other factors like reputation, our setting has no moral hazard or adverse selection, and the partnerships do not involve a large initial investment.

On the basis of the growing association between opioid addiction and HIV infection, we develop a compartmental model to study dynamics and optimal control of two epidemics; opioid addiction and HIV infection. We show that the disease-free equilibrium is locally asymptotically stable when the basic reproduction number ${ℛ}_0=\max \left({ℛ}_0^u,{ℛ}_0^v\right)<1$, here ${ℛ}_0^v$ is the reproduction number of the HIV infection, and ${ℛ}_0^u$ is the reproduction number of the opioid addiction. The addiction-only boundary equilibrium exists when ${ℛ}_0^u>1$ and it is locally asymptotically stable when the invasion number of the opioid addiction is ${ℛ}_{inv}^u<1$. Similarly, HIV-only boundary equilibrium exists when ${ℛ}_0^v>1$ and it is locally asymptotically stable when the invasion number of the HIV infection is ${ℛ}_{\text{inv}}^v<1$. We study structural identifiability of the parameters, estimate parameters employing yearly reported data from Central for Disease Control and Prevention (CDC), and practical identifiability of estimated parameters. We observe the basic reproduction number ℛ₀ using the parameters. Next, we introduce four distinct controls in the HIV — opioid model for the sake of control approach, including treatment for addictions, health care education about not sharing syringes, highly active anti-retroviral therapy (HAART), and rehab treatment for opiate addicts who are HIV infected. We apply single control in the beginning and observe the results, this provides us better understanding of the influence of each single control in the HIV–opioid model. We then apply those four controls together in the HIV–opioid model and compare their results using different control bounds and state variable weights. We conclude the result by presenting several graphs.

Iterative algorithms for solving the data assimilation problems are considered, based on the main and adjoint equations. Spectral properties of the control operators of the problem are studied, the iterative algorithms are justified.

This paper deals with maximum principles for locally Lipschitz cost functionals, under constraints given by functional equations associated with pairs of closed range unbounded operators. The so called “range condition” introduced by the author, plays an essential role. Such abstract maximum principles have both a unifying effect in this area and applications to optimal control of some partial differential equations.

In this paper we consider the coupling between one-dimensional and two-dimensional shallow water models. To this aim we introduce a domain decomposition technique with overlap to solve the coupled 1D-2D shallow water system. The proposed method is based on the introduction of suitable boundary control functions and generalizes, to the situation of heterogeneous differential problems in space, the method proposed in,¹ when the heterogeneity is confined at the differential level. We present some numerical results to assess the effectiveness of the proposed method.

Fast, dynamic or energy efficient locomotion of bipeds is still an unsolved problem in robotics. Nature seems to have solved many of the arising difficulties in thousands of years of evolution, optimizing both mechanics and control along the way. This paper proposes to use techniques from numerical optimization and optimal control combined with behavior-based control concepts to address some of the problems when designing and controlling two-legged robots. Results from the optimization process will affect both the mechanical construction and the control strategies.

We study three stochastic differential games. In each game, two players control a process X = {X_t, 0 ≤ t < ∞} which takes values in the interval I = (0,1), is absorbed at the endpoints of I, and satisfies a stochastic differential equation

In the first of our games, which is zero-sum, player 𝔄 has a continuous reward function u : [0,1] → ℝ. In addition to α(·), player 𝔄 chooses a stopping rule τ and seeks to maximize the expectation of u(X_τ); whereas player 𝔅 chooses β(·) and aims to minimize this expectation.

In the second game, players 𝔄 and 𝔅 each have continuous reward functions u(·) and v(·), choose stopping rules τ and ρ, and seek to maximize the expectations of u(X_τ) and v(X_ρ), respectively.

In the third game the two players again have continuous reward functions u(·) and v(·), now assumed to be unimodal, and choose stopping rules τ and ρ. This game terminates at the minimum τ∧ρ the stopping rules τ and ρ, and players 𝔄, 𝔅 want to maximize the expectations of u(X_τ∧ρ) and ν(X_τ∧ρ), respectively.

Under mild technical assumptions we show that the first game has a value, and find a saddle point of optimal strategies for the players. The other two games are not zero-sum, in general, and for them we construct Nash equilibria.

This paper establishes the state equation with capability of buffer and output state equation with control item for flow-shop system by Cohens max-algebra. It also proves that the sufficient and necessary condition for the non-blocking and uniform control with no buffer is a feedback of linear state. The optimal control of minimal period on the changing order set can be transferred into the calculation for problem of trip salesman, and the optimal arrangement is a special optimal control. The important parameters of the system can be expressed directly through the man-hour matrix; accordingly it is avoidable that the difficulty of calculation is caused in the analysis for the matrix by too much dimension.

A standard optimal control problem - one with general functional, nonlinear phase system and convex compact bounding set on control function, is considered. On the basis of optimization of a variation procedure and constructive use of nonclassical approximation of a cost functional, new variants of gradient methods are constructed. The quality of the proposed modifications is determined by nonlocal descent property for bilinear problems and by possibility of improvement for stationary controls in nonconvex problems.

In this article we propose a technique for deriving necessary optimality conditions and numerical optimization methods in optimal control problems of systems of partial differential equations with smooth restricted controls.

In this article we consider the generalized problem of Bolza with different time varying delays appearing in the state and velocity variables. We derive existence of solutions and prove a natural extension of a decoupling theorem, which was originally introduced by Clarke in the non delay case² and was previously extended⁷ to the neutral case when both of the delays appearing are given by one function.

An optimal control problem with linear dynamics and convex but not necessarily quadratic and possibly infinite-valued or nonsmooth costs can be analyzed in an appropriately formulated duality framework. The paper presents key elements of such a framework, including a construction of a dual optimal control problem, optimality conditions in open loop and feedback forms, and a relationship between primal and dual optimal values. Some results on differentiability and local Lipschitz continuity of the gradient of the optimal value function associated with a convex optimal control problem are included.

In this paper we consider optimal control problem for a class of infinite dimensional uncertain systems. Necessary conditions of optimality is presented under the assumption that the principal operator is monotone in a reflexive Banach space. A computational algorithm is also given.

According to the requirements of some types of missile which attack target with terminal desired impact time at the terminal point, a new guidance problem with terminal impact time constraint is investigated, which can be applied to salvo attack of anti-ship missiles. The closed form solution based on optimal control method is derived. The solution is combination of the PNG law and the feedback of the impact time error, which is difference between the designated impact time and the impact time by PNG. The new guidance law can be used to guide multiple missiles to hit a stationary simultaneously at a desirable impact time. Numerical simulation results demonstrate that the proposed guidance law can guide missiles to hit the target with desired impact time simultaneously.

Suitably applied pre-storage heat treatment (35-45°C) can maintain better fruit quality by delaying the ripening process of fruit during storage. The respiration rate of treated fruits remains lower compared with non-treated fruits during subsequent storage. In this paper, the temperature was the control input, varying from 5~40°C, and the quality changes of tomato fruit were indicated by fruit respiration rate, which was monitored as the output. The purpose was to minimize the respiration by the temperature (e.g. pre-storage heat treatment). An objective function was given by the reciprocal number of the average value of the respiration. The dynamic change in the respiration, as affected by temperature, was first identified using neural networks. The neural-network model showed good agreement with the observed responses of respiration. Optimal control of the respiration was then carried out, aiming to reduce the respiration as much as possible. For convenience, the control process was divided into 1-steps. The optimal 1-step set points of temperature which maximized the objective function were obtained using genetic algorithms. Optimal values were obtained under several different constraints of the temperature.

This paper studies optimal control policies of quality level and price for the introduction of a new product with two competing technology generations in a dynamic environment and also proposes a new model in this regard. Lots of work has been done to study the optimal policies pertaining to explanatory variables like price, promotional effort, quality, time etc. In comparison high technology products have received less attention. The proposed model is a combination of diffusion models and the cost function, which is capable of estimating the future profit trends. The new model uses the relationship between the repeat purchasers and the new purchasers in the overall diffusion of a new technology over multiple generations, by separately identifying the two types of adopters.

We consider the problem of optimal feedback control of a quantum system with linear dynamics undergoing continual non-demolition measurement of position and/or momentum, or both together. Specifically, we show that a stable domain of solutions for the filtered state of the system will be given by a class of randomized squeezed states and we exercise the control problem amongst these states. Bellman's principle is then applied directly to optimal feedback control of such dynamical systems and the Hamilton Jacobi Bellman equation for the minimum cost is derived. The situation of quadratic performance criteria is treated as the important special case and solved exactly for the class of relaxed states.

We consider a PDE-constrained optimization problem governed by a free boundary fluid-elasticity interaction. The challenge of applying optimization tools to free boundary fluid-structure interactions is the proper derivation of the adjoint sensitivity information with correct balancing conditions on the common interface. As the interaction is a coupling of Eulerian (fluid state) and Lagrangian (motion of the solid) quantities, sensitivity analysis on the system falls into the framework of shape analysis. This paper builds upon the sensitivity theory developed in^{5, 6, 8} and provides a description for the adjoint linearized model, necessary in the derivation of optimality conditions for controls and derivative-based optimization algorithms in free boundary fluid-structure interactions.

This chapter reports the application of Differential Evolution (DE) to optimize the performance of a fermentation process for xylitol production by Candida mogii yeast using a fed-batch reactor. A dynamic model that assumes the growth of cells without mortality rate was used; it has been validated earlier via experimental studies on a laboratoryscale unit. Different operating scenarios for this fermentation process have been considered, and where the impact of the decision variables has been analyzed. Numerical performance of DE algorithm has been compared with the results obtained using Particle Swarm Optimization (PSO) and Artificial Bee Colony (ABC). This comparison shows that DE algorithm is a reliable method for the dynamic optimization of xylitol production in a fed-batch fermentation process, and it can outperform both PSO and ABC algorithms.