Unbounded Control Research Articles

Stochastic Optimal Control of Quasi Integrable Hamiltonian Systems Subject to Actuator Saturation

A modified bounded optimal control strategy for quasi integrable Hamiltonian systems subject to actuator saturation is proposed. First, an n degree-of-freedom (DOF) weakly controlled quasi Hamiltonian system subject to Gaussian white noise excitation is formulated and converted into Itô equations by adding Wong-Zakai correction terms. Then, the averaged Itô stochastic differential equations and dynamical programming equation are derived by using stochastic averaging method and the stochastic dynamical programming principle, respectively. The optimal control law consisting of unbounded optimal control and bounded bang-bang control is obtained from solving the dynamical programming equation. Finally, an example of a two DOF controlled quasi integrable Hamiltonian system is worked out in detail. Numerical results show that the proposed control strategy has high control effectiveness and efficiency and the chattering is reduced significantly comparing with bang-bang control strategy.

Stochastic Target Problems with Controlled Loss

We consider the problem of finding the minimal initial data of a controlled process which guarantees to reach a controlled target with a given probability of success or, more generally, with a given level of expected loss. By suitably increasing the state space and the controls, we show that this problem can be converted into a stochastic target problem, i.e., finding the minimal initial data of a controlled process which guarantees to reach a controlled target with probability one. Unlike in the existing literature on stochastic target problems, our increased controls are valued in an unbounded set. In this paper, we provide a new derivation of the dynamic programming equation for general stochastic target problems with unbounded controls, together with the appropriate boundary conditions. These results are applied to the problem of quantile hedging in financial mathematics and are shown to recover the explicit solution of Follmer and Leukert [Finance Stoch., 3 (1999), pp. 251-273].

Numerical Approximation for a Superreplication Problem under Gamma Constraints

We study a superreplication problem of European options with gamma constraints in mathematical finance. The initially unbounded control problem is set back to a problem involving a viscosity PDE solution with a set of bounded controls. Then a numerical approach is introduced, unconditionally stable with respect to the mesh steps. A generalized finite difference scheme is used since basic finite differences cannot work in our case. Numerical tests illustrate the validity of our approach.

Stochastic optimal bounded control of MDOF quasi nonintegrable-Hamiltonian systems with actuator saturation

A new bounded optimal control strategy for multi-degree-of-freedom (MDOF) quasi nonintegrable-Hamiltonian systems with actuator saturation is proposed. First, an n-degree-of-freedom (n-DOF) controlled quasi nonintegrable-Hamiltonian system is reduced to a partially averaged Ito stochastic differential equation by using the stochastic averaging method for quasi nonintegrable-Hamiltonian systems. Then, a dynamical programming equation is established by using the stochastic dynamical programming principle, from which the optimal control law consisting of optimal unbounded control and bang–bang control is derived. Finally, the response of the optimally controlled system is predicted by solving the Fokker–Planck–Kolmogorov (FPK) equation associated with the fully averaged Ito equation. An example of two controlled nonlinearly coupled Duffing oscillators is worked out in detail. Numerical results show that the proposed control strategy has high control effectiveness and efficiency and that chattering is reduced significantly compared with the bang–bang control strategy.

Stochastic optimal control of strongly nonlinear systems under wide-band random excitation with actuator saturation

A bounded optimal control strategy for strongly non-linear systems under non-white wide-band random excitation with actuator saturation is proposed. First, the stochastic averaging method is introduced for controlled strongly non-linear systems under wide-band random excitation using generalized harmonic functions. Then, the dynamical programming equation for the saturated control problem is formulated from the partially averaged Ito equation based on the dynamical programming principle. The optimal control consisting of the unbounded optimal control and the bounded bang-bang control is determined by solving the dynamical programming equation. Finally, the response of the optimally controlled system is predicted by solving the reduced Fokker-Planck-Kolmogorov (FPK) equation associated with the completed averaged Ito equation. An example is given to illustrate the proposed control strategy. Numerical results show that the proposed control strategy has high control effectiveness and efficiency and the chattering is reduced significantly comparing with the bang-bang control strategy.

Degenerate problem of stabilization of a specified trajectory

Consideration is given to the problem of stabilization of a specified (supporting) mode as a degenerate problem of minimization of the root-mean-square deviation from it where there are no deviations of control actions. This formulation makes it possible to completely use available limited control resources for stabilization. We propose a method of approximate optimal synthesis in the neighborhood of a turnpike manifold that is obtained as a solution in the form of optimal synthesis of linearly quadratic problem with unbounded linear control. The investigation is performed for a linearized discrete model of the controlled system; however, the obtained stabilizing control can be applied in the initial nonlinear system directly or after the correction by regular iterations.

Nonlinear stochastic optimal bounded control of hysteretic systems with actuator saturation

A modified nonlinear stochastic optimal bounded control strategy for random excited hysteretic systems with actuator saturation is proposed. First, a controlled hysteretic system is converted into an equivalent nonlinear nonhysteretic stochastic system. Then, the partially averaged Ito stochastic differential equation and dynamical programming equation are established, respectively, by using the stochastic averaging method for quasi non-integrable Hamiltonian systems and stochastic dynamical programming principle, from which the optimal control law consisting of optimal unbounded control and bang-bang control is derived. Finally, the response of optimally controlled system is predicted by solving the Fokker-Planck-Kolmogorov (FPK) equation associated with the fully averaged Ito equation. Numerical results show that the proposed control strategy has high control effectiveness and efficiency.

Uniqueness results for boundary value problems arising from finite fuel and other singular and unbounded stochastic control problems

We establish uniqueness of viscosity solutions for some boundary value problems arising from stochastic optimal control problems with unbounded, possibly singular, controls. They involve a nonlinear degenerate second order Bellman-Isaacs equation and mixed boundary conditions (Dirichlet, generalized Dirichlet and state constrained conditions).

Lower Bounded Control-Lyapunov Functions

The well known Brockett condition - a topological obstruction to the existence of smooth stabilizing feedback laws - has engendered a large body of work on discontinuous feedback stabilization. The purpose of this paper is to introduce a class of control-Lyapunov from which it is possible to specify a (possibly discontinuous) stabilizing feedback law. For control-affine systems with unbounded controls Sontag has described a Lyapunov pair which gives rise to an explicit stabilizing feedback law smooth away from the origin - Sontag's universal construction of Artstein's Theorem. In this work we introduce the more general lower bounded control-Lyapunov function and a universal formula for nonaffine systems. Our universal formula is a static state feedback which is measurable and locally bounded but possibly discontinuous. Thus, for the corresponding closed loop system, the classical notion of solution need not apply. To deal with this situation we use the generalized solution due to Filippov.

Constraint-based automatic verification of abstract models of multithreaded programs

AbstractWe present a technique for the automated verification of abstract models of multithreaded programs providing fresh name generation, name mobility, and unbounded control. As high level specification language we adopt here an extension of communication finite-state machines with local variables ranging over an infinite name domain, called TDL programs. Communication machines have been proved very effective for representing communication protocols as well as for representing abstractions of multithreaded software. The verification method that we propose is based on the encoding of TDL programs into a low level language based on multiset rewriting and constraints that can be viewed as an extension of Petri Nets. By means of this encoding, the symbolic verification procedure developed for the low level language in our previous work can now be applied to TDL programs. Furthermore, the encoding allows us to isolate a decidable class of verification problems for TDL programs that still provide fresh name generation, name mobility, and unbounded control. Our syntactic restrictions are in fact defined on the internal structure of threads: In order to obtain a complete and terminating method, threads are only allowed to have at most one local variable (ranging over an infinite domain of names).

Weighted Admissibility and Wellposedness of Linear Systems in Banach Spaces

We study linear control systems in infinite‐dimensional Banach spaces governed by analytic semigroups. For $p\in[1,\infty]$ and $\alpha\in\mathbb{R}$ we introduce the notion of $L^p$‐admissibility of type $\alpha$ for unbounded observation and control operators. Generalizing earlier work by Le Merdy [J. London Math. Soc. (2), 67 (2003), pp. 715–738] and Haak and Le Merdy [Houston J. Math., 31 (2005), pp. 1153–1167], we give conditions under which $L^p$‐admissibility of type $\alpha$ is characterized by boundedness conditions which are similar to those in the well‐known Weiss conjecture. We also study $L^p$‐wellposedness of type $\alpha$ for the full system. Here we use recent ideas due to Pruss and Simonett [Arch. Math. (Basel), 82 (2004), pp. 415–431]. Our results are illustrated by a controlled heat equation with boundary control and boundary observation where we take Lebesgue and Besov spaces as state space. This extends the considerations in [C. I. Byrnes et al., J. Dynam. Control Systems, 8 (2002), pp. 341–370] to non‐Hilbertian settings and to $p\neq2$.

Non-compact-valued stochastic control under state constraints

In the present paper, we study a necessary condition under which the solutions of a stochastic differential equation governed by unbounded control processes, remain in an arbitrarily small neighborhood of a given set of constraints. We prove that, in comparison to the classical constrained control problem with bounded control processes, a further assumption on the growth of control processes is needed in order to obtain a necessary and sufficient condition in terms of viscosity solution of the associated Hamilton–Jacobi–Bellman equation. A rather general example illustrates our main result.

A stochastically averaged optimal control strategy for quasi-Hamiltonian systems with actuator saturation

A stochastic optimal control strategy for quasi-Hamiltonian systems with actuator saturation is proposed based on the stochastic averaging method and stochastic dynamical programming principle. First, the partially completed averaged Itô stochastic differential equations for the energy processes of individual degree of freedom are derived by using the stochastic averaging method for quasi-Hamiltonian systems. Then, the dynamical programming equation is established by applying the stochastic dynamical programming principle to the partially completed averaged Itô equations with a performance index. The saturated optimal control consisting of unbounded optimal control and bounded bang–bang control is determined by solving the dynamical programming equation. Numerical results show that the proposed control strategy significantly improves the control efficiency and chattering attenuation of the corresponding bang–bang control.

Conservative boundary control systems

We study continuous time linear dynamical systems of boundary control/observation type, satisfying a Green–Lagrange identity. Particular attention is paid to systems which have a well-defined dynamics both in the forward and the backward time directions. As we change the direction of time we also interchange inputs and outputs. We show that such a boundary control/observation system gives rise to a continuous time Livšic–Brodskiĭ (system) node with strictly unbounded control and observation operators. The converse is also true. We illustrate the theory by a classical example, namely, the wave equation describing the reflecting mirror.

Comparison of the bounded and unbounded feedback controls for the stochastic linear-quadratic problem

The control laws for the stochastic linear undamped system buried in Gaussian white noise were compared. Consideration was given to the magnitude-bounded and unbounded linear feedback controls. The laws were compared for the purpose of possible replacement by the linear-quadratic procedure of the procedure of determination of the switching lines for the bounded-control law.

Uniqueness Results for Second-Order Bellman--Isaacs Equations under Quadratic Growth Assumptions and Applications

In this paper, we prove a comparison result between semicontinuous viscosity sub- and supersolutions growing at most quadratically of second-order degenerate parabolic Hamilton--Jacobi--Bellman and Isaacs equations. As an application, we characterize the value function of a finite horizon stochastic control problem with unbounded controls as the unique viscosity solution of the corresponding dynamic programming equation.

The Control Transmutation Method and the Cost of Fast Controls

In this paper, the null-controllability in any positive time T of the first-order equation (1) ${\dot{x}}(t)=e^{i\theta}Ax (t)+Bu(t)$ ($|{\theta}| < \pi/2$ fixed) is deduced from the null-controllability in some positive time L of the second-order equation (2) $\ddot{z}(t)=Az(t)+Bv(t)$. The differential equations (1) and (2) are set in a Banach space, B is an admissible unbounded control operator, and A is a generator of cosine operator function. The control transmutation method makes explicit the input function u of (1) in terms of the input function v of (2): $u(t)=\int_{\mathbb{R}} k(t,s)v(s)\, ds $, where the compactly supported kernel k depends only on T and L. This method proves roughly that the norm of a u steering the system (1) from an initial state $x(0)=x_{0}$ to the final state $x(T)=0$ grows at most like $\|{x_{0}}\|\exp(\alpha_{*} L^{2}/T)$ as the control time T tends to zero. (The rate $\alpha_{*}$ is characterized independently by a one-dimensional controllability problem.) In applications to the cost of fast controls for the heat equation, L is roughly the length of the longest ray of geometric optics which does not intersect the control region.

Reachable set of bilinear control systems with time varying drift

In this article, we give a complete characterization of the reachable set at all times for a class of bilinear control systems with time varying drift and unbounded control amplitude. These results are of fundamental interest in geometric control theory and have important applications to control of coupled spins in solid state NMR spectroscopy.

Time optimal control of coupled qubits under nonstationary interactions

In this article, we give a complete characterization of all the unitary transformations that can be synthesized in a given time for a system of coupled spin-1/2 in presence of general time varying coupling tensor. Our treatment is quite general and our results help to characterize the reachable set at all times for a class of bilinear control systems with time varying drift and unbounded control amplitude. These results are of fundamental interest in geometric control theory and have applications to control of coupled spins in solid state NMR spectroscopy.

On controllability of diagonal systems with one-dimensional input space

This paper deals with diagonal systems on a Hilbert state space with a one-dimensional input space and a (possibly unbounded) control operator. A priori it is not assumed that the input operator is admissible. Necessary and sufficient conditions for different notions of controllability such as null-controllability, exact controllability and approximate controllability are presented. These conditions, which are given in terms of the eigenvalues of the diagonal operator and in terms of the control operator, are linked with the theory of interpolation in Hardy spaces.