Uniform null controllability of a linear KdV equation using two controls

In this paper we consider a linear KdV equation posed on a bounded interval. We study the behavior of the cost of null controllability when two boundary controls are employed. By means of suitable Carleman inequalities and a new exponential dissipation estimate, we prove that uniform null controllability with respect to the dispersion coefficient holds, contrary to the case when one control is used at the left end-point of the interval.     

Bang-bang principle of time optimal controls and null controllability of fractional order parabolic equations

In this paper, we establish a bang-bang principle of time optimal controls for a controlled parabolic equation of fractional order evolved in a bounded domain Ω of R^n, with a controller w to be any given nonempty open subset of Ω. The problem is reduced to a new controllability property for this equation, i.e. the null controllability of the system at any given time T 〉 0 when the control is restricted to be active in ω× E, where E is any given subset of [0, T] with positive （Legesgue） measure. The desired controllability result is established by means of a sharp observability estimate on the eigenfunctions of the Dirichlet Laplacian due to Lebeau and Robbiano, and a delicate result in the measure theory due to Lions.     

Null controllability and global blowup controllability of ordinary differential equations with feedback controls

This paper concerns the problem of feedback null controllability and blowup controllability with feedback controls for ordinary differential equations. First, we study the feedback null controllability on a time-varying ordinary differential system by unbounded feedback operators. Then, the global exact blowup controllability with feedback controls is derived on a time-invariant ordinary differential system. Finally, we obtain the approximate null controllability by bounded feedback operators, and get the approximate blowup controllability with feedback controls for ordinary differential equations.     

On a linear pursuit game with an unknown trap

A linear pursuit game with a trap, the location of which is unknown to the evader, is defined and investigated. The cases in which one of the players has complete energy dominance over his adversary are solved completely. In the general case, when no player dominates, the solution is indicated for the two-stage game.This research was supported in part by the Technion Fund for promotion of research.     

Unique winning policies for linear differential pursuit games

We study optimal control problems which are the duals, in a specified sense, to a certain class of linear differential games. Directly verifiable conditions, in terms of the data of the game, for uniqueness of solutions of the dual problem and thus for uniqueness of winning policies for the differential game, are derived. As a byproduct, in the particular context of two-dimensional problems, a strong result concerning normality is obtained. As a second byproduct, several geometrical and topological properties of thestar difference are derived. This set operation is of paramount importance for the study of rich classes of differential and difference games extending far beyond that treated here.Notation co(P) convex hull of a setPR ⁿ - ext(P) set of extreme points ofP - [x ₁,x ₂] line segment joining the pointsx ₁,x ₂ R ⁿ - S(P, c) Supporting closed halfspace ofP with exterior normalc - H(P, c) supporting hyperplane ofP with exterior normalc - F(P, c) face of the polytopeP with exterior normalc - span(P) linear span ofP - lin(P) linear closure ofP = smallest linear manifold containingP - relbd(P) relative boundary ofP - int(P) interior ofP with respect to the topology ofR ⁿ - ri(P) relative interior ofP with respect to lin(P) - a, b inner product (inR ⁿ) ofa andb - U/W {x:x U R ⁿ andx W R ⁿ} - cl(P) closure ofP This work was done during the year 1972, when the author was a student of Prof. O. Hajek at Case Western Reserve University, Cleveland, Ohio. The author wishes to thank Dr. Hajek for his many comments, suggestions, and critique.     

Equivalences between differential games and optimal controls

It is shown that linear differential pursuit games with linear targets, if player controls are required to be piecewise constant or if player controls areL ₁-functions (but pursuer control is bang-bang whenever quarry control is), are equivalent to a linear, autonomous control problem. As a byproduct, a sufficient condition for terminating the game, in Pontryagin's sense, is obtained.The present paper has been influenced by Prof. O. Hajek's work in differential games; the converse parts of the proofs presented here are very similar to those in Ref. 5. The author wishes to thank Dr. Hajek for his suggestions, comments, and critique. The Brasilian Government BNDE provided partial financial support.     

Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, II: Lack of exact controllability and controllability for very smooth solutions

This is a second paper in a two part series. In the prequel, [S.S. Krigman, C.E. Wayne, Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, I: Spectral controllability, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa2006.06.101], we showed that a system of Maxwell's equations for a homogeneous medium in a cube with nonnegative conductivity possesses the property that any finite combination of eigenfunctions is controllable (spectral controllability) by means of boundary surface currents applied over only one face of the cube. In the present paper it is established, by modifying the calculations in [H.O. Fattorini, Estimates for sequences biorthogonal to certain complex exponentials and boundary control of the wave equation, in: New Trends in Systems Analysis, Proceedings of the International Symposium, Versailles, 1976, in: Lecture Notes in Control and Inform. Sci., vol. 2, Springer, Berlin, 1977, pp. 111-124], that spectral controllability is the strongest result possible for this geometry, since the exact controllability fails regardless of the size of the conductivity term. However, we do establish controllability of solutions that are smooth enough that the Fourier coefficients of their initial data decay at an appropriate exponential rate. This does not contradict the lack of exact controllability since in any Sobolev space there are initial conditions which violate these restrictions.     

Infinite dimensional differential games with hybrid controls

A two-person zero-sum infinite dimensional differential game of infinite duration with discounted payoff involving hybrid controls is studied. The minimizing player is allowed to take continuous, switching and impulse controls whereas the maximizing player is allowed to take continuous and switching controls. By taking strategies in the sense of Elliott-Kalton, we prove the existence of value and characterize it as the unique viscosity solution of the associated system of quasi-variational inequalities.     

Global null controllability of nonlinear delay equations with controls in a compact set

The question of controlling some nonlinear retarded functional differential equations from an initial function to the zero function is considered. The control setsM() are square integrable functions with values in the unit closed sphereL ₂([t ₀, ),E ^m ) with center the origin. Assuming that the linear approximation of the nonlinear equation is null controllable with some integrable controls on some interval [t ₀,t ₁–2r], wheret ₀ is sufficiently large and wherer>0 is the delay, and assuming that the nonlinear system, withu=0, is uniformly globally asymptotically stable, we show that the nonlinear control process is globally null controllable with controlsuM(). The paper gives conditions which guarantee the stability assumptions, and also indicates conditions which yield the null controllability assumptions of the linear approximation. Our research extends known results on ordinary differential processes.     

Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, I: Spectral controllability

This is a first paper in a series of two. In both papers, we consider the question of control of Maxwell's equations in a homogeneous medium with positive conductivity by means of boundary surface currents. The domain under consideration is a cube, where the conductivity is allowed to take on any nonnegative value. An additional restriction imposed in order to make this problem more suitable for practical implementations is that the controls are applied over only one face of the cube. In this paper, the method of moments is employed to establish spectral controllability for the above case (meaning that any finite combination of eigenfunctions is controllable). In the companion paper [S.S. Krigman, C.E. Wayne, Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, II: Lack of exact controllability and controllability for very smooth solutions, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa2006.02.102] it will be established, by modifying the calculations in [H.O. Fattorini, Estimates for sequences biorthogonal to certain complex exponentials and boundary control of the wave equation, in: New Trends in Systems Analysis, Proceedings of the International Symposium, Versailles, 1976, in: Lecture Notes in Control and Inform. Sci., vol. 2, Springer, Berlin, 1977, pp. 111-124], that exact controllability fails for this geometry regardless of the size of the conductivity term. However, we will also establish in [S.S. Krigman, C.E. Wayne, Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, II: Lack of exact controllability and controllability for very smooth solutions, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa2006.02.102] controllability of solutions that are smooth enough that the Fourier coefficients of their initial data decay at a suitable exponential rate.     

A class of stochastic games with ordered field property

It is shown that discounted general-sum stochastic games with two players, two states, and one player controlling the rewards have the ordered field property. For the zero-sum case, this result implies that, when starting with rational data, also the value is rational and that the extreme optimal stationary strategies are composed of rational components.     

Remarks on the controllability of fractional differential equations

In this paper, we study the approximate controllability of a linear fractional differential equation by using the method of regularization of Tikhonov. New concepts and results about controllability are established. Then, under the condition of the positivity of the controllability operator, we obtain that the linear system can be steered to an arbitrary small neighbourhood of the fractional integral of the state at final time.     

Linear controllability by piecewise constant controls with assigned switching times

For a linear time-dependent control process inR ⁿ, we prove that the complete controllability by means of the space of all the admissible controls is equivalent to the complete controllability by means of a suitablen-dimensional space of piecewise constant controls with at mostn preassigned switching times. An analogous result is also established for more general controllability problems.     

Discrete-time zero-sum Markov games with first passage criteria

In this paper, we deal with two-person zero-sum stochastic games for discrete-time Markov processes. The optimality criterion to be studied is the discounted payoff criterion during a first passage time to some target set, where the discount factor is state-dependent. The state and action spaces are all Borel spaces, and the payoff functions are allowed to be unbounded. Under the suitable conditions, we first establish the optimality equation. Then, using dynamic programming techniques, we obtain the existence of the value of the game and a pair of optimal stationary policies. Moreover, we present the exponential convergence of the value iteration and a ‘martingale characterization’ of a pair of optimal policies. Finally, we illustrate the applications of our main results with an inventory system.     

Feedback strategies with finite memory in differential games

It is shown that, in differential games, strategies can be defined for the players in such a way that their controls depend, for each timet, on a finite section of the past of the trajectoryx(t). In particular,s-delay,r-memory strategies can be defined as in the Varaiya-Lin approach. It is shown that, for deterministic differential games with terminal payoff, the upper and lower values of the game so defined, are independent of the lengthr of the memory. Lettingr 0, a feedback strategy is obtained which depends only on the present (and infinitesimal past) of the trajectoryx(t).     

Relative approximate controllability of abstract functional systems with infinite delay and delayed control

This paper is concerned with the relative approximate controllability of functional systems with infinite delay and delayed control in Hilbert spaces. In particular, we begin with studying some criteria of the controllability of linear systems. Based on those results, sufficient conditions are derived for the relative approximate controllability of nonlinear functional systems. Finally, an example is included to illustrate the effectiveness of the proposed methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite-time null controllability for a class of linear evolution equations on a Banach space with control constraints

We present some necessary and sufficient conditions for null controllability for a class of general linear evolution equations on a Banach space with constraints on the control space. We also present a result on the existence of time-optimal controls and some partial results on the maximum principle. Some interesting insights that can be obtained from these results are discussed, and the paper is concluded with an application to a boundary control problem.This work was supported in part by the National Science and Engineering Council of Canada under Grant No. 7109.The author is thankful to Professor L. Cesari for many helpful suggestions and also for calling his attention to the recent papers of Professor K. Narukawa.     

On stochastic games with additive reward and transition structure

In this paper, we introduce a new class of two-person stochastic games with nice properties. For games in this class, the payoffs as well as the transitions in each state consist of a part which depends only on the action of the first player and a part dependent only on the action of the second player.For the zero-sum games in this class, we prove that the orderfield property holds in the infinite-horizon case and that there exist optimal pure stationary strategies for the discounted as well as the undiscounted payoff criterion. For both criteria also, finite algorithms are given to solve the game. An example shows that, for nonzero sum games in this class, there are not necessarily pure stationary equilibria. But, if such a game possesses a stationary equilibrium point, then there also exists a stationary equilibrium point which uses in each state at most two pure actions for each player.     

Zero-sum games with incompletely defined payoff

It is shown that a saddle-point solution exists in a two-person, zero-sum game whose payoff is given by a matrix which is not completely defined. On the other hand, we show that such games do not always have a value, so that a saddle-point solution is not necessarily an optimal solution.This work was supported by the Centre d'Etudes Atomiques, Saclay, France.