Necessary Conditions For Optimality Research Articles

Optimal Persistent Monitoring Using Second-Order Agents With Physical Constraints

This paper addresses a one-dimensional optimal persistent monitoring problem using second-order agents. The goal is to control the movements of agents to minimize a performance metric associated with the environment (targets) over a finite time horizon. In contrast to earlier results limited to first-order dynamics for agents, we control their accelerations rather than velocities, thus leading to a better approximation of agent behavior in practice and to smoother trajectories. Bounds on both velocities and accelerations are also taken into consideration. Despite these added complications to agent dynamics, we derive a necessary condition for optimality and show that the optimal agent trajectories can be fully characterized by two parameter vectors. A gradient-based algorithm is proposed to optimize these parameters and yield a minimal performance metric. In addition, a collision avoidance algorithm is proposed to solve potential collision and boundary-crossing problems, thus extending the gradient-based algorithm solutions. Finally, simulation examples are included to demonstrate the effectiveness of our results.

Investigation of one linear non-smooth two-parameter discrete optimal control problem

The object of research is the linear optimal control problem described by discrete two-parameter systems under the assumption that the controlled process is stepwise.The work is aimed at deriving the necessary first-order optimality conditions in the case of a non-smooth quality function. And also to establish the necessary conditions of second-order optimality in stepwise control problems for discrete two-parameter systems. The paper investigates one linear two-parameter discrete optimal control problem with a non-smooth quality criterion. A special increment of the quality functional is calculated. Cases under the condition of a convex set are considered. The concept of special control in the problem under study is given. A number of necessary optimality conditions for the first and second orders are established. And also the necessary second-order optimality conditions are obtained in terms of directional derivatives. In the case of a linear quality criterion, the necessary and sufficient optimality condition is proved using the increment formula by analogous arguments. Under the assumption that the set is convex, a special increment of the quality criterion for admissible control is defined.The methods of calculus of variations and optimal control, the theory of difference equations are used. The result is obtained for the optimality of a special, first-order control, in the case of convexity of the set. The case when the minimized functional is linear is considered. In this case, a necessary and sufficient condition is obtained for the optimality of the admissible control.Thanks to the research results, it is possible to obtain the necessary first-order optimality conditions in terms of directional derivatives in the stepwise problem of optimal control of discrete two-parameter systems. As well as the necessary conditions of optimality of the second order in the case of convexity of the control domain and the necessary optimality conditions of special controls.The theoretical results obtained in the work are of interest in the theory of optimal control of step systems and can be used in the further development of the theory of necessary optimality conditions for step control problems.

Gradient method using pseudospectral collocation scheme for two-stage optimal control with an unspecified switching time

A two-stage gradient method (TGM) based on traditional backward integration is firstly proposed for solving two-stage nonlinear optimal control problem with arbitrary performance index and unspecified switching time. To reduce the computational cost, pseudospectral collocation scheme is embedded so as to transfer a family of time-varying influence function dynamics into a set of linear algebraic equations. Therefore, a series of symbolic formulation can be analytically derived to efficiently calculate the influence functions. Also, some special integrals using to update the improvements can be efficiently determined by Gaussian Quadrature. Finally, the performance of both TGMs is investigated with reduced-order reentry trajectory planning problem. The results show that both methods only take a few steps to find the solution satisfying the necessary condition for optimality and terminal constraints. Furthermore, the TGM with pseudospectral collocation scheme considerably improves the computational efficiency and has the potential to be applicable in the framework of real-time guidance.

Discrete time optimal control with frequency constraints for non-smooth systems

We present a Pontryagin maximum principle for discrete time optimal control problems with (a) pointwise constraints on the control actions and the states, (b) smooth frequency constraints on the control and the state trajectories, and (c) nonsmooth dynamical systems. Pointwise constraints on the states and the control actions represent desired and/or physical limitations on the states and the control values; such constraints are important and are widely present in the optimal control literature. Constraints of type (b), while less standard in the literature, effectively serve the purpose of describing important properties of inertial actuators and systems. The conjunction of constraints of type (a) and (b) is a relatively new phenomenon in optimal control but is important for the synthesis control trajectories with a high degree of fidelity. The maximum principle established here provides first order necessary conditions for optimality that serve as a starting point for the synthesis of control trajectories corresponding to a large class of constrained motion planning problems that have high accuracy in a computationally tractable fashion. Moreover, the ability to handle a reasonably large class of nonsmooth dynamical systems that arise in practice ensures broad applicability of our theory.

BER-Improved Quantization of Source-to-Relay Link SNR for Cooperative Beamforming: A Fixed Point Theory Approach

Cooperative beamforming is a potentially useful technique for enhancing link reliability in modern wireless relay communications. To aid the beamforming design, the signal-to-noise ratio (SNR) of the source-to-relay (S-R) channel links must be known at the destination node. Abdallah and Papadopoulos in IEEE Trans. Signal Processing , vol. 56, no. 10, 2008 , proposed a beamforming system with relay-assisted SNR acquisition; that is, the S-R link SNR is first quantized at each relay and is then forwarded to the destination. An optimal quantizer design problem was proposed in that paper, aiming at reducing the system bit error rate; however, only the binary quantization case was considered therein, with the quantization threshold computed by a suggested rule of thumb. In this paper, we study the aforementioned quantizer design problem in the general multiple-bit setting. We show that the solutions to the first-order necessary condition for optimality can be obtained as a fixed point of a certain nonlinear map over the feasible threshold set. The existence and uniqueness of the fixed point are established by using, respectively, the Brouwer’s and Schauder’s fixed point theorems. Finally, we show that the fixed point thus obtained yields the globally optimal set of quantization thresholds. To the best of our knowledge, this paper is the first which leverages fixed point theories to rigorously solve SNR quantization problems in the context of wireless relay communication.

Least-squares solution of a class of optimal space guidance problems via Theory of Connections

In this paper, we apply a newly developed method to solve boundary value problems for differential equations to solve optimal space guidance problems in a fast and accurate fashion. The method relies on the least-squares solution of differential equations via orthogonal polynomials expansion and constrained expression as derived via Theory of Connection (ToC). The application of the optimal control theory to derive the first order necessary conditions for optimality, yields a Two-Point Boundary Value Problem (TPBVP) that must be solved to find state and costate. Combining orthogonal polynomials expansion and ToC, we solve the TPBVP for a class of optimal guidance problems including energy-optimal landing on planetary bodies and fixed-time optimal intercept for a target-interceptor scenario. The performance analysis in terms of accuracy shows the potential of the proposed methodology as applied to optimal guidance problems.

Numerical Solution for Classical Optimal Control Problem Governing by Hyperbolic Partial Differential Equation via Galerkin Finite Element-Implicit method with Gradient Projection Method

This paper deals with the numerical solution of the discrete classical optimal control problem (DCOCP) governing by linear hyperbolic boundary value problem (LHBVP). The method which is used here consists of: the GFEIM " the Galerkin finite element method in space variable with the implicit finite difference method in time variable" to find the solution of the discrete state equation (DSE) and the solution of its corresponding discrete adjoint equation, where a discrete classical control (DCC) is given. The gradient projection method with either the Armijo method (GPARM) or with the optimal method (GPOSM) is used to solve the minimization problem which is obtained from the necessary condition for optimality of the DCOCP to find the DCC.An algorithm is given and a computer program is coded using the above methods to find the numerical solution of the DCOCP with step length of space variable , and step length of time variable . Illustration examples are given to explain the efficiency of these methods. The results show the methods which are used here are better than those obtained when we used the Gradient method (GM) or Frank Wolfe method (FWM) with Armijo step search method to solve the minimization problem.

Sensitivity of Optimal Control Problems Arising from their Hamiltonian Structure

First-order necessary conditions for optimality reveal the Hamiltonian nature of optimal control problems. Regardless of the overwhelming awareness of this result, the implications that it entails have not been fully explored. We discuss how the symplectic structure of optimal control constrains the flow of sub-volumes in the phase space. Special emphasis is devoted to dynamics in the neighborhood of optimal trajectories and insight is gained into how errors in the initial states affect terminal conditions. Specifically, we prove that if the optimal trajectory does not satisfy a particular condition, then there exists a set of variations in the initial states yielding a greater error in norm when mapped to the terminal time through the state transition matrix. We relate this result to the sensitivity problem in solving indirect problems for optimal control.

Optimal Control for Partially Observed Nonlinear Interval Systems.

In this paper, we consider the optimal control problem for a class of systems governed by nonlinear time-varying partially observed interval differential equations. The control process is assumed to be governed by linear time varying interval differential equation driven by the observed process. Using the fact that the state, observation, and control processes possess lower and upper bounds, we have developed sets of (ordinary) differential equations that describe the behavior of the bounds of these processes. Using these differential equations, the interval control problem can be transformed into an equivalent ordinary control problem in which interval mathematics and extension principle of Moore are not required. Using variational arguments, we have developed the necessary conditions of optimality for the equivalent (ordinary) control problem. Finally, we present some numerical simulations to illustrate the effectiveness of the proposed control scheme.

Dynamic interpolation for obstacle avoidance on Riemannian manifolds

ABSTRACT This work is devoted to studying dynamic interpolation for obstacle avoidance. This is a problem that consists of minimising a suitable energy functional among a set of admissible curves subject to some interpolation conditions. The given energy functional depends on velocity, covariant acceleration and on artificial potential functions used for avoiding obstacles. We derive first-order necessary conditions for optimality in the proposed problem; that is, given interpolation and boundary conditions we find the set of differential equations describing the evolution of a curve that satisfies the prescribed boundary values, interpolates the given points and is an extremal for the energy functional. We study the problem in different settings including a general one on a Riemannian manifold and a more specific one on a Lie group endowed with a left-invariant metric. We also consider a sub-Riemannian problem.

Role of Media and Effects of Infodemics and Escapes in the Spatial Spread of Epidemics: A Stochastic Multi-Region Model with Optimal Control Approach

Mass vaccination campaigns play major roles in the war against epidemics. Such prevention strategies cannot always reach their goals significantly without the help of media and awareness campaigns used to prevent contacts between susceptible and infected people. Feelings of fear, infodemics, and misconception could lead to some fluctuations of such policies. In addition to the vaccination strategy, the movement restriction approach is essential because of the factor of mobility or travel. However, anti-epidemic border measures may also be disturbed if some infected travelers manage to escape and infiltrate into a safer region. In this paper, we aim to study infection dynamics related to the spatial spread of an epidemic in interconnected regions in the presence of random perturbations caused by the three above-mentioned reasons. Therefore, we devise a stochastic multi-region epidemic model in which contacts between susceptible and infected populations, vaccination-based and movement restriction optimal control approaches are all assumed to be unpredictable, and then, we discuss the effectiveness of such policies. In order to reach our goal, we employ a stochastic maximum principle version for noised systems, state and prove the sufficient and necessary conditions of optimality, and finally provide the numerical results obtained using a stochastic progressive-regressive schemes method.

A geometric approach for the optimal control of difference inclusions

Difference inclusions provide a discrete-time analogue of differential inclusions, which in turn play an important role in the theories of optimal control, implicit differential equations, and invariance and viability, to name a few. In this paper we: (i) introduce a framework suitable for the study of difference inclusions for which the state evolves on a manifold; (ii) use this framework to develop necessary conditions for optimality for a broad class of discrete-time problems of dynamic optimization in which the state evolves on a manifold M. The necessary conditions for optimality we derive include the case for which the state $$q_i$$ is subject to constraints $$q_i \in S_i \subseteq M$$ , for $$S_i$$ a closed set. The resulting necessary conditions for optimality appear as discrete-time versions of the Euler–Lagrange inclusion studied by Ioffe (in Trans Am Math Soc 349(7):2871–2900, 1997), Ioffe and Rockafellar (in Calc Var Partial Differ Equ 4(1):59–87, 1996), Mordukhovich (in SIAM J Control Optim 33(3):882–915, 1995), Mordukhovich (in Variational analysis and generalized differentiation II: applications. Springer, Berlin, 2006), and Vinter and Zheng (in SIAM J Control Optim 35(1):56–77, 1997) generalized in a natural way to the case in which the state is evolving on a manifold.

Application of Optimal Control Theory to Newcastle Disease Dynamics in Village Chicken by Considering Wild Birds as Reservoir of Disease Virus

In this study, an optimal control theory was applied to a nonautonomous model for Newcastle disease transmission in the village chicken population. A notable feature of this model is the inclusion of environment contamination and wild birds, which act as reservoirs of the disease virus. Vaccination, culling, and environmental hygiene and sanitation time dependent control strategies were adopted in the proposed model. This study proved the existence of an optimal control solution, and the necessary conditions for optimality were determined using Pontryagin’s Maximum Principle. The numerical simulations of the optimal control problem were performed using the forward–backward sweep method. The results showed that the use of only the environmental hygiene and sanitation control strategy has no significant effect on the transmission dynamics of the Newcastle disease. Additionally, the combination of vaccination and environmental hygiene and sanitation strategies reduces more number of infected chickens and the concentration of the Newcastle disease virus in the environment than any other combination of control strategies. Furthermore, a cost-effective analysis was performed using the incremental cost-effectiveness ratio method, and the results showed that the use of vaccination alone as the control measure is less costly compared to other control strategies. Hence, the most effective way to minimize the transmission rate of the Newcastle disease and the operational costs is concluded to be the timely vaccination of the entire population of the village chicken, improvement in the sanitation of facilities, and the maintenance of a hygienically clean environment.

Necessary Conditions for Optimality in Problems of Optimal Control of Systems with Discontinuous Right-Hand Side

We consider the problems of optimal control of a dynamical system whose right-hand side is discontinuous in the state variable and is linear in the control with sufficiently smooth coefficients in each of the half-spaces into which the space is divided by the switching hyperplane. The main attention is paid to the situation where there exist intervals on which the optimal trajectory lies on the switching surface. New nondegenerate necessary conditions for optimality are stated and proved in the maximum principle form. The obtained optimality conditions are compared with the already known conditions.

Closed-form solution of optimal control problem of a fractional order system

A formulation and an analytical solution technique of fractional optimal control problem (FOCP) is presented in this paper. The performance index (PI) considered is a conformable fractional integral (CFI) function and is a function of both the state and the control variables. Dynamic behaviour of the system is described by conformable fractional differential equation (FDE). The necessary conditions of optimality and the general transversality condition in terms of Hamiltonian are obtained using variational approach. Both the fixed and free final end point conditions have been considered. An analytical solution technique is presented for solving the conformable FOCPs. To validate the formulation and solution scheme numerical examples are presented.

Optimal low-thrust hyperbolic rendezvous for Earth-Mars missions

Earth-Mars cycling spacecraft have been proposed as a valuable option for the future exploration of Mars. Cycler mission architectures consider the use of a large space vehicle that cycles continuously between Earth and Mars, describing a near-ballistic path that includes flybys at the two planets. While this large spacecraft can be equipped with the life support system appropriate for a long interplanetary flight with a crew, taxi vehicles of reduced size are sufficient to ensure the connection between the interplanetary vehicle and each planet. This research addresses the determination of the optimal low-thrust transfer trajectories for a pair of taxi vehicles aimed at transferring a crew or an inert payload from and toward a large Earth-Mars cycling spacecraft. The problem is solved through the indirect heuristic method, which uses the necessary conditions for optimality together with a heuristic technique, i.e. the particle swarm algorithm. The use of the indirect heuristic approach for low-thrust paths may be affected by hypersensitivity with respect to the initial values of the costate. This phenomenon is mitigated through a judicious choice of the equations of motion that govern the spacecraft dynamics. The solution method proposed in this work proves to be effective and very accurate, and leads to determining the minimum-time low-thrust paths for a pair of taxi vehicles traveling from and toward a large cycling spacecraft.

The maximum principle for discrete-time control systems and applications to dynamic games

We study deterministic nonstationary discrete-time optimal control problems in both finite and infinite horizon. With the aid of Gâteaux differentials, we prove a discrete-time maximum principle in analogy with the well-known continuous-time maximum principle. We show that this maximum principle, together a transversality condition, is a necessary condition for optimality; we also show that it is sufficient under additional hypothesis. We use Gâteaux differentials as a natural setting to derive first-order conditions. Additionally, we use the discrete-time maximum principle to derive the discrete-time Euler equation and to characterize Nash equilibria for discrete-time dynamic games.

Modeling, simulation and optimal control strategy for batch fermentation processes

The use of fermenters at large scale is usually hampered by sub-optimal conditions in terms of yield and productivity, along with the low tolerance of strains to process stresses, such as substrate and product toxicity, and other fermentation inhibitors. Attempts to improve the industrial efficacy of fermenters have been in the areas of genetic engineering to improve strain tolerance, but this usually involves detailed and unfeasible mechanistic studies. Statistical designs of experiments have also been used to optimize industrial fermenters but this again often results in local optima due to the relatively small-dimensional space covered by the experiments. Mathematical techniques have recorded great successes and regarding ethanol fermentation with sorghum extracts, previous work has modeled and established the presence of product inhibition, however, did not consider other degrees of freedom (temperature and pH) that minimize the effect of such inhibitions. This paper includes the description of a batch alcohol fermentation process that has been optimized using a technique based on the application of mathematical modeling and optimal control. Calculus of variation is introduced as a valuable tool to derive and solve the necessary conditions for optimality, and the obtained results show the optimal temperature and pH profiles for the fermentation of sorghum extracts. A Simulink model of the fermentation process shows that using the proposed control strategy increases ethanol yield by 14.18%, cell growth by 71.96% decreases the residual substrate by 84.77%.

Necessary First-Order and Second-Order Optimality Conditions in Discrete-Time Stochastic Systems

In this paper, first-order and second-order necessary conditions for optimality for discrete-time stochastic optimal control problems governed by discrete-time Ito equations are established. A new discrete-time backward stochastic equation and discrete-time backward stochastic matrix equation are introduced. Based on the discrete-time backward stochastic Ito equation, a discrete-time stochastic maximum principle for the stochastic discrete optimal control problems is obtained. Moreover, using the discrete-time backward stochastic matrix equation the second-order necessary conditions for optimality are obtained.

Self-optimizing control methodology for mixed integer programming problems: a case study of refinery production scheduling

Problem formulation as mixed integer nonlinear programming (MINLP) is one of the most challenging task in refinery scheduling optimization. In most of the work reported in refinery scheduling, uncertainties from design point of view predominate. However, there is also a need to consider operational uncertainties (disturbances) as they affect the accuracy and robustness of the overall schedule. This study proposed a novel approach under self- optimizing control (SOC) framework to deal with multi-period refinery scheduling problems under uncertain conditions. The goal is to maintain global optimum by controlling the gradient of the cost function at zero via approximating necessary conditions of optimality (NCO) over the whole uncertain parameter space. A regression model for the plant expected revenue (profit) as a function of independent variables using optimal operation data was obtained and a feedback input (manipulated variable) was derived. The performance of the proposed approach was tested using case studies. The first case assumed a system with no disturbance with the base case model giving an optimal profit of $56,696,407 while the proposed approach yields $50,523,054, translating to 10.888 % loss. The percentage loss for the second, third and fourth cases with disturbances are 5.807 %, 4.409% and 7.898% respectively. The results obtained have shown that the idea presented was able to effectively deal with the situation at hand with percentage loss within a reasonable degreeKeywords: Refinery scheduling scheduling, MINLP formulation, Operational uncertainty (disturbances), Necessary condition of optimality, Feedback control