On the Order Reduction Approach for Singularly Perturbed Optimal Control Systems

The order reduction method for singularly perturbed optimal control systems consists of employing the system obtained while setting the small parameter to be zero. In many situations the differential-algebraic system thus obtained indeed provides an appropriate approximation to the singularly perturbed problem with a small parameter. In this paper we establish that if relaxed controls are allowed then the answer to the question whether or not this method is valid depends essentially on one simple parameter: the dimension of the fast variable, denoted n. More specifically, if n=1 then the order reduction method is indeed applicable, while if n>1 then the set of singularly perturbed optimal control systems for which it is not applicable is dense (in the L norm).     

Emergence of time-asymptotic flocking for a general Cucker–Smale-type model with distributed time delays

In this paper, we analyze the asymptotic flocking behavior for a Cucker–Smale-type model with a disturbed delayed coupling, where delays are information processing and reactions of individuals. By constructing a new energy functional combined with L²-analysis, we obtain the uniform bound of particle velocities, and then by establishing a system of dissipative differential inequalities together with L^∞-analysis, we prove the existence of asymptotic flocking solutions when the maximum value of time delays is sufficiently small.     

Quantum Optimal Control Problems with a Sparsity Cost Functional

In this article, the investigation of a class of quantum optimal control problems with L¹ sparsity cost functionals is presented. The focus is on quantum systems modeled by Schrödinger-type equations with a bilinear control structure as it appears in many applications in nuclear magnetic resonance spectroscopy, quantum imaging, quantum computing, and in chemical and photochemical processes. In these problems, the choice of L¹ control spaces promotes sparse optimal control functions that are conveniently produced by laboratory pulse shapers. The characterization of L¹ quantum optimal controls and an efficient numerical semi-smooth Newton solution procedure are discussed.     

Global strong solutions and time decay of 2D tropical climate model with zero thermal diffusion

The global small solutions of the tropical climate model are obtained with the fractional dissipative terms Λ^αu in the equation of the barotropic mode u and Λ^αv in the equation of the first baroclinic mode v. More precisely, we prove for 1<α ≤ 2 that the couple system has global unique strong solutions for small initial data with critical regularities. Moreover, the smallness assumption imposed on the initial barotropic mode of the velocity can be removed if α=2. We also study the large time behavior of the constructed solutions and obtain optimal time decay rates by a pure energy argument.     

Numerical solution of a nonlinear time-optimal control problem

Nonlinear systems with a stationary (i.e., explicitly time independent) right-hand side are considered. For time-optimal control problems with such systems, an iterative method is proposed that is a generalization of one used to solve nonlinear time-optimal control problems for systems divided by phase states and controls. The method is based on constructing finite sequences of simplices with their vertices lying on the boundaries of attainability domains. Assuming that the system is controllable, it is proved that the minimizing sequence converges to an ɛ-optimal solution after a finite number of iterations. A pair {T, u(·)} is called an ɛ-optimal solution if |T − T _opt| − ɛ, where T _opt is the optimal time required for moving the system from the initial state to the origin and u is an admissible control that moves the system to an ɛ-neighborhood of the origin over the time T.     

Near-optimal mean–variance controls under two-time-scale formulations and applications

Although the mean–variance control was initially formulated for financial portfolio management problems in which one wants to maximize the expected return and control the risk, our motivations stem from highway vehicle platoon controls that aim to maximize highway utility while ensuring zero accident. This paper develops near-optimal mean–variance controls of switching diffusion systems. To reduce the computational complexity, with motivations from earlier work on singularly perturbed Markovian systems [Sethi and Zhang, Hierarchical Decision Making in Stochastic Manufacturing Systems, Birkhäuser, Boston, MA, 1994; Yin and Zhang, Continuous-Time Markov Chains and Applications: A Singular Pertubation Approach, Springer-Verlag, New York, 1998 and Yin et al., Ann. Appl. Probab. 10 (2000), pp. 549–572], we use a two-time-scale formulation to treat the underlying system, which is represented by the use of a small parameter. As the small parameter goes to 0, we obtain a limit problem. Using the limit problem as a guide, we construct controls for the original problem, and show that the control so constructed is nearly optimal.     

Riccati equations for generalized singular estimate control systems

We consider the Bolza problem associated with boundary/point control systems governed by strongly continuous semigroups. In continuation of our work in Lasiecka and Tuffaha [I. Lasiecka and A. Tuffaha, Riccati equations for the Bolza problem arising in boundary/point control problems governed by C ₀–semigroups satisfying a singular estimate, J. Optim. Theory Appl. 136 (2008), pp. 229–246; I. Lasiecka and A. Tuffaha, A Bolza optimal synthesis problem for singular estimate control systems, Control Cybernet 38(4B) (2009), pp. 1429–1460], we yet extend the theory to a more general class of control problems that are not analytic providing sharp blow-up rates for the regularity. Solvability of the associated Riccati equations and an optimal feedback synthesis are established. The presence of unbounded control actions, such as boundary/point controls, naturally lead to a singularity at the terminal point t?=?T of the optimal control and of the corresponding feedback operator as before. The class of control systems considered in this article is a generalization to the class usually referred to in the literature as ‘Singular Estimate Control Systems’. The prototype is still that of a PDE system consisting of coupled hyperbolic parabolic dynamics interacting on an interface with point/boundary control. The distinct feature of the class considered in this article is that the degree of unboundedness in the control is stronger than that allowed in the usual singular estimate control system configuration, giving rise to less regular optimal state trajectories.     

Global existence and time decay rates for the 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosities

We consider the Cauchy problem for the three-dimensional bipolar compressible Navier-Stokes-Poisson system with unequal viscosities.Under the assumption that the H₃ norm of the initial data is small but its higher order derivatives can be arbitrarily large,the global existence and uniqueness of smooth solutions are obtained by an ingenious energy method.Moreover,if additionally,the H^?s(1/2≤s<3/2)or B^?s_2,∞(1/2相似文献   

On the optimal controls of a class of systems governed by second order parabolic partial delay-differential equations with first boundary conditions

Summary In this paper we consider the question of existence of optimal controls for a class of systems governed by second order parabolic partial delay-differential equations with first boundary conditions and with controls appearing in the coefficients. In Theorems2.2 and2.3 we present existence and uniqueness of solutions of the first boundary problems. In Theorems3.1 and3.2 we prove that whenever the coefficients of the system converge in the w*-topology (L¹ topology on L^∞) the corresponding solutions converge weakly in an appropriate Sobolev space. Using these basic results we present two theorems (Theorems4.1 and4.2) on the existence of optimal controls. Entrata in Redazione il 21 gennaio 1978.     

Variation formulas of solutions and optimal control problems for differential equations with retarded argument

The authors prove a theorem on the continuous dependence of solutions of nonlinear systems of differential equations with variable delay on the perturbations of initial data (initial instant, initial function, and initial value of the trajectory) and the right-hand side in the case where these perturbations are small in the Euclidean and integral topology, respectively. The variation formulas of solutions of a differential equation with discontinuous and continuous initial condition are deduced; as compared with those known earlier, these formulas take into account the variation of the initial instant and the discontinuity and continuity of the initial data. A necessary condition for criticality of mappings defined on a finitely locally convex set is obtained. The quasiconvexity of filters in studying optimal problems with delays in controls is proved. Necessary optimality conditions and existence theorems are proved for optimal problems with variable delays in phase coordinates and controls having a nonfixed initial instant, a discontinuous and a continuous initial condition, and functional and boundary conditions of general form. Necessary optimality conditions are obtained for optimal problems with variable structure and delays. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 25, Optimal Control, 2005.     

Optimal control problems for linear fractional-order systems defined by equations with Hadamard derivative

Two optimal control problems are studied for linear stationary systems of fractional order with lumped variables whose dynamics is described by equations with Hadamard derivative, a minimum-norm control problem and a time-optimal problem with a constraint on the norm of the control. The setting of the problem with nonlocal initial conditions is considered. Admissible controls are sought in the class of functions p-integrable on an interval for some p. The main approach to the study is based on the moment method. The well-posedness and solvability of the moment problem are substantiated. For several special cases, the optimal control problems under consideration are solved analytically. An analogy between the obtained results and known results for systems of integer and fractional order described by equations with Caputo and Riemann–Liouville derivatives is specified.

     

Time optimal factorizations on compact Lie groups

In this paper we study the relationship between factorization problems on SU(2ⁿ) or more generally on compact Lie groups G and time optimal control problems. Both types of problems naturally arise in physics, such as in quantum computing and in controlling coupled spin systems (NMR‐spectroscopy). In the first part we show that certain factorization problems can be reformulated as time optimal control problems on G. In the second part a necessary condition for the existence of finite optimal factorizations is discussed. At the end we illustrate our results by an example on Euler angle factorizations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of Optimal Controls for Partially Observed Jump Processes

A partially observable control problem for an R ^d -valued jump process with counting observations is studied. The state and the observations may be strongly dependent and, in particular, the two processes may jump together. An equivalent separated problem is introduced and the existence of an optimal control for the separated problem is obtained in the class of relaxed and generalized controls. Equivalence between the initial problem and the relaxed generalized separated control problem is discussed.     

Ergodicity and Lyapunov Functions for Langevin Dynamics with Singular Potentials

We study Langevin dynamics of N particles on ℝ^d interacting through a singular repulsive potential, e.g., the well-known Lennard-Jones type, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted total variation distance. The proof of the main result relies on an explicit construction of a Lyapunov function. In contrast to previous results for such systems, our result implies geometric convergence to equilibrium starting from an essentially optimal family of initial distributions. © 2019 Wiley Periodicals, Inc.     

Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method

In this article one discusses the controllability of a semi-discrete system obtained by discretizing in space the linear 1-D wave equation with a boundary control at one extremity. It is known that the semi-discrete models obtained with finite difference or the classical finite element method are not uniformly controllable as the discretization parameter h goes to zero (see [8]). Here we introduce a new semi-discrete model based on a mixed finite element method with two different basis functions for the position and velocity. We show that the controls obtained with these semi-discrete systems can be chosen uniformly bounded in L²(0,T) and in such a way that they converge to the HUM control of the continuous wave equation, i.e. the minimal L²-norm control. We illustrate the mathematical results with several numerical experiments. Supported by Grant BFM 2002-03345 of MCYT (Spain) and the TMR projects of the EU ``Homogenization and Multiple Scales" and ``New materials, adaptive systems and their nonlinearities: modelling, control and numerical simulations". Partially Supported by Grant BFM 2002-03345 of MCYT (Spain), Grant 17 of Egide-Brancusi Program and Grant 80/2005 of CNCSIS (Romania).     

On time-minimal distributed control of vibrating systems governed by an abstract wave equation

Distributed control of vibrations governed by an abstract wave equation is studied. First it is shown that every initial state of finite energy can be transferred to a position of rest within any finite time interval by a unique control with minimumL ²-norm. If only controls with a uniformly boundedL ²-norm are admitted, the same statement is shown for sufficiently large time intervals. In this case the existence of time-minimal null-controls can be proved by routine arguments. In addition, time-minimal controls are characterized by the property of being least norm controls on the minimum time interval and having asL ²-norm exactly the upper bound that is prescribed. The results partly overlap with results of Fattorini.     

Existence of optimal controls for systems governed by parabolic partial differential equations with Cauchy boundary conditions

Summary This paper considers the existence of optimal controls for systems governed by a second order parabolic partial differential equation in divergence form with Cauchy conditions. As preliminary results, theorems concerning the convergence of the sequence of weak solutions corresponding to a sequence of admissible controls are proved. Two general forms of criteria are considered. The first one is taken as a function of the weak solution of the system, and the other is taken as a function of the solution of the system and control. Several theorems and corollaries on the existence of optimal controls are then presented.     

Optimal attitude control of a rigid body

Being mainly interested in the control of satellites, we investigate the problem of maneuvering a rigid body from a given initial attitude to a desired final attitude at a specified end time in such a way that a cost functional measuring the overall angular velocity is minimized.This problem is solved by applying a recent technique of Jurdjevic in geometric control theory. Essentially, this technique is just the classical calculus of variations approach to optimal control problems without control constraints, but formulated for control problems on arbitrary manifolds and presented in coordinate-free language. We model the state evolution as a differential equation on the nonlinear state spaceG=SO(3), thereby completely circumventing the inevitable difficulties (singularities and ambiguities) associated with the use of parameters such as Euler angles or quaternions. The angular velocities _k about the body's principal axes are used as (unbounded) control variables. Applying Pontryagin's Maximum Principle, we lift any optimal trajectorytg^*(t) to a trajectory onT ^*G which is then revealed as an integral curve of a certain time-invariant Hamiltonian vector field. Next, the calculus of Poisson brackets is applied to derive a system of differential equations for the optimal angular velocitiest _k ^* (t); once these are known the controlling torques which need to be applied are determined by Euler's equations.In special cases an analytical solution in closed form can be obtained. In general, the unknown initial values _k ^* (t₀) can be found by a shooting procedure which is numerically much less delicate than the straightforward transformation of the optimization problem into a two-point boundary-value problem. In fact, our approach completely avoids the explicit introduction of costate (or adjoint) variables and yields a differential equation for the control variables rather than one for the adjoint variables. This has the consequence that only variables with a clear physical significance (namely angular velocities) are involved for which gooda priori estimates of the initial values are available.     

Uniform Decay properties of a model in structural acoustics

We investigate decay properties for a system of coupled partial differential equations which model the interaction between acoustic waves in a cavity and the walls of the cavity. In this system a wave equation is coupled to a structurally damped plate or beam equation. The underlying semigroup for this system is not uniformly stable, but when the system is appropriately restricted we obtain some uniform stability. We present two results of this type. For the first result, we assume that the initial wave data is zero, and the initial plate or beam data is in the natural energy space; then the corresponding solution to system decays uniformly to zero. For the second result, we assume that the initial condition is in the natural energy space and the control function is L²(0,∞) (in time) into the control space; then the beam displacement and velocity are both L²(0,∞) into a space with two spatial derivatives.     

Periodic optimal control for parabolic Volterra-Lotka type equations

This paper considers the optimal harvesting control of a biological species, whose growth is governed by the parabolic diffusive Volterra-Lotka equation. We prove that such equation with L^∞ periodic coefficients has an unique positive periodic solution. We show the existence and uniqueness of an optimal control, and under certain conditions, we characterize the optimal control in terms of a parabolic optimality system. A monotone sequence which converges to the optimal control is constructed.