State constrained optimal control problems with time delays

In a recent, related, paper, necessary conditions in the form of a Maximum Principle were derived for optimal control problems with time delays in both state and control variables. Different versions of the necessary conditions covered fixed end-time problems and, under additional hypotheses, free end-time problems. These conditions improved on previous conditions in the following respects. They provided the first fully non-smooth Pontryagin Maximum Principle for problems involving delays in both state and control variables, only special cases of which were previously available. They provide a strong version of the Weierstrass condition for general problems with possibly non-commensurate control delays, whereas the earlier literature does so only under structural assumptions about the dynamic constraint. They also provided a new ‘two-sided’ generalized transversality condition, associated with the optimal end-time. This paper provides an extension of the Pontryagin Maximum Principle of the earlier paper for time delay systems, to allow for the presence of a unilateral state constraint. The new results fully recover the necessary conditions of the earlier paper when the state constraint is absent, and therefore retain all their advantages but in a setting of greater generality.     

Regularization of optimal design problems for bars and plates,part 2

For the problems of optimal design considered in Ref. 1, contradictions arising in the necessary conditions of optimality are eliminated by suitable extension of the initially given class of admissible materials. The extended class includes composites of some special (layered) microstructure. Elastic properties of such composites are described, and alternative (regularized) formulations of the optimal design problems are given. Necessary conditions of Weierstrass are shown to be satisfied, both for the case in which the strip of variations is small compared with the width of the layers and for the opposite case. Numerical results are given for the regularized problem of a bar of extremal torsional rigidity.The authors are indebted to Dr. N. A. Lavrov for performing numerical calculations.     

Optimal Control of a Variational Inequality with Application to Equilibrium Problem of an Elastic Nonhomogeneous and Anisotropic Plate Resting on Unilateral Elastic Foundation

Several optimal control problems with the same state problem—a variational inequality with a monotone operator—are considered. The inequality represents bending of an elastic, nonhomogeneous, anisotropic Kirchhoff plate resting on some unilateral elasto-rigid foundation and point supports. Both the thickness of the plate and the coefficient of the unilateral elastic foundation play the role of design variables. Cost functionals include the work of external forces (compliance), total reaction forces of the foundation or the weight of the plate. The solvability of all the problems is proved. Moreover, approximate methods for the optimal control and weight minimization problems are proposed, making use of finite elements. The design variables are approximated by piecewise affine functions. The solvability of the approximate problems is proved and some convergence analysis is presented.     

Discontinuities in hyperbolic optimal problems

Optimal problems are discussed for systems governed by hyperbolic equations in cases where the characteristic (phase) velocities depend upon the control functions. It is assumed that the phase velocity depends upon both spatial coordinates and time.The problems in question lead to certain difficulties connected with the formulation of the Weierstrass necessary condition for a minimum. Actually, strong discontinuities may arise in their solutions when we perform avariation in a strip. These discontinuities provide additional possibilities which are useful in certain minimization problems.In the first part of the paper, the case of a single quasilinear equation of the first order is discussed; an optimal problem for one-dimensional wave equation is described in the second part. In both cases, it turns out that additional information is needed, and usually this information is provided by physical arguments. This results of the paper can be generalized to the case where the number of spatial coordinates exceeds one.The author is indebted to Dr. K. G. Guderley for his very helpful comments.     

Optimal switching in coplanar orbit transfer

The set of all controls that satisfy the Weierstrass necessary condition for optimality in the problem of time-open, coplanar orbit transfer via impulses is presented, along with the switching relations that must be satisfied at a corner in an optimal trajectory. This includes detailed data for eccentricities near unity. This study takes advantage of recently discovered closed-form solutions for the switching surfaces of this problem.Portions of this work were supported by NASA Contract No. NASr-54(06) and by NASA Grant No. NGR-06-003-033.     

Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions

A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre–Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh interval. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre–Gauss–Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Furthermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.

     

On the optimal distribution of elastic moduli of a nonhomogeneous body

The problem of the optimal distribution of elastic moduli is considered for a linearly clastic inhomogeneous body. The cost function is taken to be the work produced by the surface tractions. Necessary conditions for stationary behavior and the Weierstrass condition are obtained. The difference between maximum and minimum problems is underlined, and pecularities connected with different cost functions are discussed.     

Analysis and optimization in problems of cloaking of material bodies for the Maxwell equations

We consider control problems for the 3D Maxwell equations describing electromagnetic wave scattering in an unbounded inhomogeneous medium that contains a permeable isotropic obstacle with cloaking boundary. Such problems arise when studying cloaking problems by the optimization method. The boundary coefficient occurring in the impedance boundary condition plays the role of a control. We study the solvability of the control problem and derive optimality systems that describe necessary conditions for the extremum. By analyzing the constructed optimality systems, we justify sufficient conditions imposed on the input data providing the uniqueness and stability of optimal solutions.     

Orientational Design of Anisotropic Materials Using the Hill and Tsai–Wu Strength Criteria

Linearly elastic two- and three-dimensional orthotropic materials are considered. The problems of optimal material orientation are studied in the cases of the Hill and Tsai–Wu strength criteria. The necessary optimality conditions are derived for a 3D orthotropic material. In the case of a 2D orthotropic material, an analytical solution is obtained. An analysis of global and local extrema is presented.     

Maximum principle in the problem of time optimal response with nonsmooth constraints : PMM vol. 40, n 6, 1976, pp. 1014–1023

The problem of optimal response [1, 2] with nonsmooth (generally speaking, nonfunctional) constraints imposed on the state variables is considered. This problem is used to illustrate the method of proving the necessary conditions of optimality in the problems of optimal control with phase constraints, based on constructive approximation of the initial problem with constraints by a sequence of problems of optimal control with constraint-free state variables. The variational analysis of the approximating problems is carried out by means of a purely algebraic method involving the formulas for the incremental growth of a functional [3, 4] and the theorems of separability of convex sets is not used.Using a passage to the limit, the convergence of the approximating problems to the initial problem with constraints is proved, and for general assumptions the necessary conditions of optimality resembling the Pontriagin maximum principle [1] are derived for the generalized solutions of the initial problem. The conditions of transversality are expressed, in the case of nonsmooth (nonfunctional) constraints by a novel concept of a cone conjugate to an arbitrary closed set of a finite-dimensional space. The concept generalizes the usual notions of the normal and the normal cone for the cases of smooth and convex manifolds.     

Two Error Bounds for Constrained Optimization Problems and Their Applications

This paper presents a global error bound for the projected gradient and a local error bound for the distance from a feasible solution to the optimal solution set of a nonlinear programming problem by using some characteristic quantities such as value function, trust region radius etc., which are appeared in the trust region method. As applications of these error bounds, we obtain sufficient conditions under which a sequence of feasible solutions converges to a stationary point or to an optimal solution, respectively, and a necessary and sufficient condition under which a sequence of feasible solutions converges to a Kuhn–Tucker point. Other applications involve finite termination of a sequence of feasible solutions. For general optimization problems, when the optimal solution set is generalized non-degenerate or gives generalized weak sharp minima, we give a necessary and sufficient condition for a sequence of feasible solutions to terminate finitely at a Kuhn–Tucker point, and a  sufficient condition which guarantees that a sequence of feasible solutions terminates finitely at a stationary point. This research was supported by the National Natural Science Foundation of China (10571106) and CityU Strategic Research Grant.     

Variational problems in the theory of optimum processes

This paper presents a review of the optimization problems for control processes described by ordinary differential equations and of the variational methods for solving these problems. The following cases are studied: problems with constraints on the controls or the coordinates, problems described by equations with discontinuous right-hand sides, problems with functionals depending on intermediate coordinates, and problems with given discontinuities in the coordinates. Variational problems of synthesis of optimal systems are also discussed. The method of solution is based on the multiplier rule and the Weierstrass necessary condition for the strong minimum of a functional. In some cases, the Legendre-Clebsch necessary condition for the weak minimum of a functional is used.     

具有算子与有界约束的无穷维规划问题的最优性条件

本文给出了当all-at-once方法用于求解最优控制问题而产生的一类同时具有算子和对部分变量具有简单界约束的无穷维最优化问题的一个一阶必要条件,构造了相应的信赖域子问题,据此,信赖域法可以用于求解无穷空间中的最优化问题。     

A second-order condition that strengthens Pontryagin's maximum principle

We derive a second-order necessary condition for optimal control problems defined by ordinary differential equations with endpoint restrictions. This condition, based on a second-order restricted minimization test, bears a somewhat similar relation to the Weierstrass E-condition (the Pontryagin maximum principle) as the Legendre and Jacobi conditions bear to the Euler-Lagrange equation. Specifically, in the context of relaxed controls, the E-condition for free endpoint problems asserts that if a function achieves its minimum over a convex set Q at some point $\text{q}$ then its one-sided directional derivatives at $\text{q}$ into Q are nonnegative. Our new condition, when applied to the special case of free endpoint problems, corresponds to the observation that if such a one-sided directional derivative at $\text{q}$ is 0 then the corresponding second directional derivative is nonnegative. This new condition effectively supplements the Pontryagin maximum principle over the singular regimes of “weakly” normal extremals that are candidates for either a relaxed or an ordinary restricted minimum. Like some other second-order methods, this condition is global over the control set but, unlike the other tests, it is also global over time. A number of examples illustrate its use and behavior.     

On the approximation of infinite optimization problems with an application to optimal control problems

This paper is concerned with approximations to infinite optimization problems in Banach spaces. Under the assumption of a first order necessary and a second order sufficient optimality condition we derive convergence results for the optimal solutions and the optimal values of the approximating problems. An application to finite difference approximations of nonlinear optimal control problems with state constraints is given.     

Control of boundary impedance in two-dimensional material-body cloaking by the wave flow method

Control problems are considered for a two-dimensional electromagnetic field model describing electromagnetic wave scattering in a unbounded homogeneous medium containing an anisotropic permeable inclusion with a partially covered (cloaked) boundary. The control is a function involved in the impedance boundary condition on the covered part of the boundary. The solvability of the original mixed transmission problem for the two-dimensional Helmholtz equation and of the control problems is proved. Optimality systems describing necessary extremum conditions are derived. The uniqueness and stability of optimal solutions with respect to certain perturbations of the cost functional and the incident wave are established.     

Duality in infinite dimensional linear programming

We consider the class of linear programs with infinitely many variables and constraints having the property that every constraint contains at most finitely many variables while every variable appears in at most finitely many constraints. Examples include production planning and equipment replacement over an infinite horizon. We form the natural dual linear programming problem and prove strong duality under a transversality condition that dual prices are asymptotically zero. That is, we show, under this transversality condition, that optimal solutions are attained in both primal and dual problems and their optimal values are equal. The transversality condition, and hence strong duality, is established for an infinite horizon production planning problem.This material is based on work supported by the National Science Foundation under Grant No. ECS-8700836.     

Robust optimal control of stochastic hyperelastic materials

Soft robots are highly nonlinear systems made of deformable materials such as elastomers, fluids and other soft matter, that often exhibit intrinsic uncertainty in their elastic responses under large strains due to microstructural inhomogeneity. These sources of uncertainty might cause a change in the dynamics of the system leading to a significant degree of complexity in its controllability. This issue poses theoretical and numerical challenges in the emerging field of optimal control of stochastic hyperelasticity. This paper states and solves the robust averaged control in stochastic hyperelasticity where the underlying state system corresponds to the minimization of a stochastic polyconvex strain energy function. Two bio-inspired optimal control problems under material uncertainty are addressed. The expected value of the L²-norm to a given target configuration is minimized to reduce the sensitivity of the spatial configuration to variations in the material parameters. The existence of optimal solutions for the robust averaged control problem is proved. Then the problem is solved numerically by using a gradient-based method. Two numerical experiments illustrate both the performance of the proposed method to ensure the robustness of the system and the significant differences that may occur when uncertainty is incorporated in this type of control problems.     

Stretching of an Elastoplastic Plate with an Arbitrarily Oriented Crack

The problem of determination of the crack resistance of an elastoplastic plate, weakened by a rectilinear slit in the form of a cut, under the conditions of uniaxial stretching is considered. The material of the body is assumed to be incompressible, reinforcing, and obeying the Mises condition of plasticity. The straining theory of plasticity is used. The solutions are obtained in the elastic and plastic regions in the form of asymptotic expansions in the neighborhood of the end of the crack. Based on the conditions that the crack borders are unloaded and the elastic and plastic solutions are conjugate on the contour of the plastic region, unknown constants of integration are found. The two-leafed contours of the plastic region are obtained. The critical load is determined.     

Variational and optimal control problems with delayed argument

This paper deals with variational and optimal control problems with delayed argument and presents analogs of the classical necessary conditions for optimality for problems in (n + 1)-space. It is mainly concerned with the functional $$J(y) = \int_a^b {f[t,y(t\user2{--}\tau ),y(t),\dot y(t\user2{--}\tau ),\dot y(t)] dt} $$ There are no side conditions; τ is a positive real number; andy is a continuous piecewise smooth vector function havingn components. The fundamental lemma of the calculus of variations is used in deriving an analog of the Euler equations. The usual construction is utilized in obtaining analogs of the Weierstrass and Legendre conditions. Also found is a fourth necessary condition involving the least proper value associated with a boundary value problem related to the second variation. A sufficient condition is obtained by the use of a simple expansion method. The last station of the paper outlines an extension of a maximal principle obtained by Hestenes to control problems which involve delays in both the state variable and the control variable.