Necessary Conditions For Optimality Research Articles

Minimum Energy Control of a Unicycle Model Robot

Abstract Point-to-point path planning for a kinematic model of a differential-drive wheeled mobile robot (WMR) with the goal of minimizing input energy is the focus of this work. An optimal control problem is formulated to determine the necessary conditions for optimality and the resulting two point boundary value problem is solved in closed form using Jacobi elliptic functions. The resulting nonlinear programming problem is solved for two variables and the results are compared to the traditional shooting method to illustrate that the Jacobi elliptic functions parameterize the exact profile of the optimal trajectory. A set of terminal constraints which lie on a circle in the first quadrant are used to generate a set of optimal solutions. It is noted that for maneuvers where the angle of the vector connecting the initial and terminal point is greater than a threshold, which is a function of the radius of the terminal constraint circle, the robot initially moves into the third quadrant before terminating in the first quadrant. The minimum energy solution is compared to two other optimal control formulations: (1) an extension of the Dubins vehicle model where the constant linear velocity of the robot is optimized for and (2) a simple turn and move solution, both of whose optimal paths lie entirely in the first quadrant. Experimental results are used to validate the optimal trajectories of the differential-drive robot.

Quantile-based optimal portfolio selection

In this paper the concept of quantile-based optimal portfolio selection is introduced and a specific portfolio connected to it, the conditional value-of-return (CVoR) portfolio, is proposed. The CVoR is defined as the mean excess return or the conditional value-at-risk (CVaR) of the return distribution. The portfolio selection consists solely of quantile-based risk and return measures. Financial institutions that work in the context of Basel 4 use CVaR as a risk measure. In this regulatory framework sufficient and necessary conditions for optimality of the CVoR portfolio are provided under a general distributional assumption. Moreover, it is shown that the CVoR portfolio is mean-variance efficient when the returns are assumed to follow an elliptically contoured distribution. Under this assumption the closed-form expression for the weights and characteristics of the CVoR portfolio are obtained. Finally, the introduced methods are illustrated in an empirical study based on monthly data of returns on stocks included in the S&P index. It is shown that the new portfolio selection strategy outperforms several alternatives in terms of the final investor wealth.

Analysis and Optimal Velocity Control of a Stochastic Convective Cahn\u2013Hilliard Equation

A Cahn–Hilliard equation with stochastic multiplicative noise and a random convection term is considered. The model describes isothermal phase-separation occurring in a moving fluid, and accounts for the randomness appearing at the microscopic level both in the phase-separation itself and in the flow-inducing process. The call for a random component in the convection term stems naturally from applications, as the fluid’s stirring procedure is usually caused by mechanical or magnetic devices. Well-posedness of the state system is addressed, and optimisation of a standard tracking type cost with respect to the velocity control is then studied. Existence of optimal controls is proved, and the Gâteaux–Fréchet differentiability of the control-to-state map is shown. Lastly, the corresponding adjoint backward problem is analysed, and the first-order necessary conditions for optimality are derived in terms of a variational inequality involving the intrinsic adjoint variables.

Optimal control strategies of a fractional order model for Zika virus infection involving various transmissions

According to the world health organization (WHO), more than 17% of infectious diseases represented in vector-borne diseases (VBDs). So, more than 700,000 deaths annually caused by the spread of these diseases. Basically, VBDs are a usually used term that characterizes a disease transmitted to a host population, by vectors like mosquitoes, insects and ticks through feeding activity. Being a vector means that it carries a disease from an infective to a healthy host. In this paper, we address the fractional mathematical model which describes the transmission dynamics for one of the VBDs, that is Zika virus. We incorporated various combinations between four controls to reduce the transmission of this disease, the measures taken to reduce the contact among vectors and humans such as asleep under a bed net, wearing full clothes, stay in places with a screen window to keep the vector outside; the efforts to reduce the transmission rate from vectors to humans by increasing the autoimmunity; the efforts made to change sexual habits among a susceptible and infected human by using a condom at the period of Zika virus outbreak; and the measures taken to increase the death rate of vectors by using the insecticide spraying. The stability of all equilibrium points (EPs) of the fractional order model (FOM) is investigated. The control reproduction number R c of the FOM is computed and the effect of the suggested controls on the behavior of R c are presented graphically. Moreover, we characterized a fractional optimal control problem (FOCP) as well as used Pontryagin’s maximum principle to derive a fractional order necessary optimality conditions (NOCs). We solved these NOCs numerically by developing the forward-backward sweep method (FBSM) using the predictor-corrector method (PCM), where the state and adjoint equations in the form of the left Caputo fractional derivative (CFD). Furthermore, the outcomes are discussed with some figures and the approximate values for the cost functional are given based on the proposed strategies. Comparing all strategies with the uncontrolled case in order to choose the best strategy for reducing the spread of Zika virus infection.

Optimal test allocation

A health authority chooses a binary action for each of several individuals that differ in their pretest probabilities of being infectious and in the additive losses associated with two types of decision errors. The authority is endowed with a portfolio of tests that differ in their sensitivities and specificities. We derive a simple necessary condition for optimality of test allocation. In special cases, precision parameters of the allocated test are monotone in the individuals' types. We characterize the marginal benefit of a test, provide an algorithmic solution for the test-allocation problem and consider the benefits of confirmatory testing.

A General Stochastic Maximum Principle for Mean-Field Controls with Regime Switching

Focusing on regime-switching diffusions with mean-fields interactions, this paper is devoted to obtaining a general maximum principle. A main feature is that conditional mean-field is used in the controlled dynamic systems and the optimization process. Analysis of variational and adjoint equations using forward and backward stochastic differential equations with regime-switching and conditional mean-field is carried out. Necessary conditions for optimality are obtained without assuming the convexity of the control space. An example on conditional mean-variance portfolio selection with regime switching is given to illustrate the sufficient conditions for optimality.

Optimal parking management of connected autonomous vehicles: A control-theoretic approach

In this paper we develop a continuous-time stochastic dynamic model for the optimal parking management of connected autonomous vehicles (CAVs) in the presence of multiple parking lots within a given area. Inspired by the well-known Lotka-Volterra equations, a mathematical model is developed to explicitly incorporate the interactions among those parking garages under consideration. Time-dependent parking space availability is considered as the system state, while the dynamic price of parking is naturally used as the control input which can be properly chosen by the parking garage operators from the admissible set. By regulating parking rates, the total demand for parking can be distributed among the set of parking lots under question. Further, we formulate an optimal control problem (called Bolza problem) with the objective of maintaining the availability (managing the demand) of each parking garage at a desired level, which could potentially reduce traffic congestion as well as fuel consumption of CAVs. Based on the necessary conditions of optimality given by Pontryagin’s minimum principle (PMP), we develop a computational algorithm to address the nonlinear optimization problem and formally prove its convergence. A series of Monte Carlo simulations is conducted under various scenarios and the corresponding optimization problems are solved determining the optimal pricing policy for each parking lot. Since the stochastic dynamic model is general and the control inputs, i.e., parking rates, are easy to implement, it is believed that the procedures presented here will shed light on the parking management of CAVs in the near future.

Necessary optimality conditions for minimax optimal control problems with mixed constraints

A weak maximal principle for minimax optimal control problems with mixed state-control equality and inequality constraints is provided. In the formulation of the minimax control problem we allow for parameter uncertainties in all functions involved: in the cost function, in the dynamical control system and in the equality and inequality constraints. Then a new constraint qualification of Mangassarian-Fromovitz type is introduced which allowed us to prove the necessary conditions of optimality. We also derived the optimality conditions under a full rank conditions type and showed that it is, as usual, a particular case of the Mangassarian-Fromovitz type condition case. Illustrative examples are presented.

Minimax Control Problems: Optimality Conditions

The formulation of the minimax control problem is considered. We allow for parameter uncertainties in all functions involved: in the cost function, in the dynamical control system and in the equality and inequality constraints. Weak maximal principle for minimax optimal control problems with mixed state-control equality and inequality constraints, under the constraint qualification of Mangassarian-Fromovitz type are discussed. The optimality conditions under a full rank conditions type is derived, and it is shown that it is, as usual, a particular case of the Mangassarian-Fromovitz type condition case. We also look at the necessary conditions of optimality for simplified versions of the model and degeneracy issues.

An efficient numerical solution for time switching optimal control problems

In this paper, an efficient computational algorithm for the solution of Hamiltonian boundary value problems arising from bang-bang optimal control problems is presented. For this purpose, at first, based on the Pontryagin’s minimum principle, the first order necessary conditions of optimality are derived. Then, an indirect shooting method with control parameterization, in which the control function is replaced with a piecewise constant function with values and switching points taken as unknown parameters, is presented. Thereby, the problem is converted to the solution of the shooting equation, in which the values of the control function and the switching points as well the initial values of the costate variables are unknown parameters. The important advantages of this method are that, the obtained solution satisfies the first order optimality conditions, further the switching points can be captured accurately which is led to an accurate solution of the bangbang problem. However, solving the shooting equation is nearly impossible without a very good initial guess. So, in order to cope with the difficulty of the initial guess, a homotopic approach is combined with the presented method. Consequently, no priori assumptions are made on the optimal control structure and number of the switching points, and sensitivity to the initial guess for the unknown parameters is resolved too. Illustrative examples are included at the end and efficiency of the method is reported.

ОПТИМАЛЬНОЕ ПОДАВЛЕНИЕ ПОПЕРЕЧНЫХ КОЛЕБАНИЙ ВРАЩАЮЩИХСЯ УПРУГИХ СТЕРЖНЕЙ

The process of suppressing transverse vibrations of an elastic rod spinning in a horizontal plane and fixed at one of its ends is studied. It is supposed that the rod spins around the vertical axis at a constant angular velocity and performs transverse vibrations in the vertical plane, the vibrations are assumed small in amplitude. Transverse vibrations of the spinning rod are performed under external mechanical action. Lateral vibrations are described by the displacement function and considered in a rotating plane by using the classical beam model. The necessary conditions of optimality are derived and applied for suppressing elastic vibrations on a finite time-interval. The problem of optimal suppression of lateral vibrations caused by initial disturbances is formulated as a variational problem with constraints that take into account the suppressing effects on the rod. The limiting restrictions are presented in the form of inequalities. With the introduction of an additional variable, these restrictions are reduced to standard integral equality, while taking into account the energy constraints imposed on the control actions. The proposed iterative algorithm for solving the formulated problem is a numerical-analytical algorithm and consists in minimizing the quadratic quality criterion. This criterion characterizes the vibration suppression process and allows the implementation of improving variations. As a result of the operations carried out, the dependence of the vibration suppression process on the determining parameters, such as the angular velocity of rotation, the isoperimetric energy constant, and the length of the considered process of vibration suppression in time, has been clarified. An example that illustrates the implementation of the proposed algorithm and shows the effectiveness of this method for suppressing lateral vibrations is given.

Rate Constrained Discrete-time Maximum Principle

Abstract Actuators in real-world systems operate under limited bandwidth and saturation constraints. Mathematically, these limitations manifest as certain types of constraints being imposed on the sequence of control action. While saturation constraints have been studied extensively, bandwidth limitations in the actuators have received very little attention in the control community. While an attempt has been made in the direction of incorporating frequency constraints at the synthesis stage, rate constraints on the control at the design stage has not been studied extensively in the regime of discrete-time. This article tries to fill in that gap by incorporating rate constraints on the control trajectories. We propose a new discrete-time Pontryagin maximum principle (PMP) with rate constraints being imposed on the control and derive a set of first order necessary conditions for optimality. As an illustration, the linear quadratic (LQ) optimal control problem under rate constraints has been considered.

Bifurcation preserving discretisations of optimal control problems

The first order optimality conditions of optimal control problems (OCPs) can be regarded as boundary value problems for Hamiltonian systems. Variational or symplectic discretisation methods are classically known for their excellent long term behaviour. As boundary value problems are posed on intervals of fixed, moderate length, it is not immediately clear whether methods can profit from structure preservation in this context. When parameters are present, solutions can undergo bifurcations, for instance, two solutions can merge and annihilate one another as parameters are varied. We will show that generic bifurcations of an OCP are preserved under discretisation when the OCP is either directly discretised to a discrete OCP (direct method) or translated into a Hamiltonian boundary value problem using first order necessary conditions of optimality which is then solved using a symplectic integrator (indirect method). Moreover, certain bifurcations break when a non-symplectic scheme is used. The general phenomenon is illustrated on the example of a cut locus of an ellipsoid.

Optimal Control with Nonregular Mixed Constraints: An Optimization Approach

We derive necessary conditions for optimality for control problems with smooth mixed constraints in the absence of regularity conditions. We show that the lack of regularity is reflected by the appearance of purely finitely additive set functions as multipliers corresponding to the mixed constraints. We illustrate with examples the rule of such set functions. Additionally, we show how our approach allows to treat regular mixed constraints and pure state constraints as particular cases; for the latter, we also provide a connection to well-established results involving only countably additive measures.

Optimal age control of bio-population dynamics

The object of this paper is the problem of optimal control of a semi-linear system of two first-order hyperbolic equations with integral constraints. Using the method of characteristics and the maximum principle the necessary conditions for optimality are obtained.

A process transfer model-based optimal compensation control strategy for batch process using just-in-time learning and trust region method

The advantages of maximally transferring similar process data for modeling make the process transfer model attract increasing attention in quality prediction and optimal control. Unfortunately, due to the difference between similar processes and the uncertainty of data-driven model, there are usually a more serious mismatch between the process transfer model and the actual process, which may result in the deterioration of process transfer model-based control strategies. In this research, a process transfer model based optimal compensation control strategy using just-in-time learning and trust region method is proposed to cope with this problem for batch processes. First, a novel JITL-JYKPLS (Just-in-time learning Joint-Y kernel partial least squares) model combining the JYKPLS (Joint-Y kernel partial least squares) process transfer model and just-in-time learning is proposed and employed to obtain the satisfactory approximation in a local region with the assistance of sufficient similar process data. Then, this paper integrates JITL-JYKPLS model with the trust region method to further compensate for the NCO (necessary condition of optimality) mismatch in the batch-to-batch optimization problem, and the problem of estimating experimental gradients is also avoided. Meanwhile, a more elaborate model update scheme is designed to supplement the lack of new data and gradually eliminate the adverse effects of partial differences between similar process production processes. Finally, the feasibility of the proposed optimal compensation control strategy is demonstrated through a simulated cobalt oxalate synthesis process.

First order necessary conditions of optimality for the two dimensional tidal dynamics system

<p style='text-indent:20px;'>In this work, we consider the two dimensional tidal dynamics equations in a bounded domain and address some optimal control problems like total energy minimization, minimization of dissipation of energy of the flow, etc. We also examine an another interesting control problem which is similar to that of the data assimilation problems in meteorology of obtaining unknown initial data, when the system under consideration is the tidal dynamics, using optimal control techniques. For these cases, different distributed optimal control problems are formulated as the minimization of suitable cost functionals subject to the controlled two dimensional tidal dynamics system. The existence of an optimal control as well as the first order necessary conditions of optimality for such systems are established and the optimal control is characterized via the adjoint variable. We also establish the uniqueness of optimal control in small time interval.</p>

Measure-Driven Nonlinear Dynamic Systems with Applications to Optimal Impulsive Controls

In this paper, we consider a class of nonlinear systems driven by measures generalizing the class of impulsive systems. We use measures as control and prove existence of optimal controls (measures) and present necessary conditions of optimality. We apply the general results to derive necessary conditions of optimality for purely impulse-driven systems. These results are then applied to optimal control problems related to geosynchronous satellites with some numerical results.

A composite framework for co-optimization of spacecraft trajectory and propulsion system

Indirect optimization methods convert optimal control problems (OCPs) into two- or multi-point boundary-value problems. A highly desirable feature of indirect methods, specifically for space applications, is that high-resolution trajectories can be generated, which satisfy the first-order necessary conditions of optimality. A recently developed Composite Smoothing Control (CSC) framework is utilized to formulate and solve the problem of simultaneous trajectory optimization and propulsion sub-system design of spacecraft. A reasonable parameterized breakdown of the spacecraft mass is adopted, which captures the impact of power produced by the solar arrays and its contribution to the total spacecraft mass. Thus, the implicit trade-offs can be considered in the indirect optimization approach. The function space co-optimization problem of spacecraft power subsystem parameters along with the main trajectory is solved with the objective to maximize the payload delivered. The proposed framework amounts to an invariant embedding that reduces the original, difficult-to-solve, multi-point boundary-value problem into a two-point boundary-value problem with continuous, differentiable control inputs. Utility of the proposed construct is demonstrated through a low-thrust, multi-revolution, multi-year rendezvous maneuver to asteroid Dionysus with a variable-specific-impulse, variable-thrust modeled engine. This is the first time that indirect optimization methods have tackled such a complex co-optimization problem using the CSC framework.

Optimality criteria method in 2D linearized elasticity problems

In optimal design problems the goal is to find an optimal distribution of materials in a fixed domain which optimizes a criterion, computed through the solution of a partial differential equation modelling the involved physics. We consider optimal design problems in the setting of linearized elasticity, and restrict ourselves to domains filled with two isotropic elastic phases. Since a classical solution usually does not exist, we use the homogenization method in order to get a proper relaxation of the original problem. Unfortunately, the related G-closure problem in linearized elasticity is not solved. However, in the compliance minimization problem, G-closure can be replaced by a smaller subset made of sequential laminates, which is explicitly known. Moreover, for the compliance minimization the necessary conditions of optimality are easily derived, which enables a development of optimality criteria method for the numerical solution. We explicitly calculate the lower Hashin-Shtrikman bound on the complementary energy in two space dimensions, and necessary conditions of optimality. We develop an optimality criteria method for the single state compliance minimization problems.