arXiv subjects

Getachew K. Befekadu

Publications and source records attributed to Getachew K. Befekadu.

32 records · Page 2Linked to original sources

arXiv2016

On the controlled eigenvalue problem for stochastically perturbed multi-channel systems

In this brief paper, we consider the problem of minimizing the asymptotic exit rate of diffusion processes from an open connected bounded set pertaining to a multi-channel system with small random perturbations. Specifically, we establish a connection between: (i) the existence of an invariant set for the unperturbed multi-channel system w.r.t. certain class of state-feedback controllers; and (ii) the asymptotic behavior of the principal eigenvalues and the solutions of the Hamilton-Jacobi-Bellman (HJB) equations corresponding to a family of singularly perturbed elliptic operators. Finally, we provide a sufficient condition for the existence of a Pareto equilibrium (i.e., a set of optimal exit rates w.r.t. each of input channels) for the HJB equations -- where the latter correspond to a family of nonlinear controlled eigenvalue problems.

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arXiv2015

Remarks on the hierarchical control problems with model uncertainty

In this paper, we consider a hierarchical control problem with model uncertainty. Specifically, we consider the following objectives that we would like to accomplish. The first one being of a controllability-type that consists of guaranteeing the terminal state to reach a target set starting from an initial condition, while the second one is keeping the state trajectory of the system close to a given reference trajectory over a finite time interval. We introduce the following framework. First, we partition the control subdomain into two disjoint open subdomains, with smooth boundaries, that are compatible with the strategy subspaces of the {\it leader} (which is responsible for the controllability-type criterion) and that of the {\it follower} (which is associated with the second criterion), respectively. Moreover, we account at the optimization stage for model uncertainty by allowing the {\it leader} to choose its control strategy based on a class of alternative models about the system, whereas the {\it follower} makes use of an approximate model about the system. Using the notion of Stackelberg's optimization, we provide conditions on the existence of optimal control strategies for such a hierarchical control problem, under which the {\it follower} is required to respond optimally to the strategy of the {\it leader} so as to achieve the overall objectives. Apart from the issue of modeling and uncertainty, this paper is a companion to our previous work.

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arXiv2015

On the attainable distributions of controlled-diffusion processes pertaining to a chain of distributed systems

We consider a controlled-diffusion process pertaining to a chain of distributed systems with random perturbations that satisfies a weak Hörmander type condition. In particular, we consider a stochastic control problem with the following objectives that we would like to achieve. The first one being of a reachability-type that consists of determining a set of attainable distributions at a given time starting from an initial distribution, while the second one involves minimizing the relative entropy subject to the initial and desired final attainable distributions. Using the logarithmic transformations approach from Fleming, we provide a sufficient condition on the existence of an optimal admissible control for such a stochastic control problem which is amounted to changing the drift by a certain perturbation suggested by Jamison in the context of reciprocal processes. Moreover, such a perturbation coincides with a minimum energy control among all admissible controls forcing the controlled-diffusion process to the desired final attainable distribution starting from the initial distribution. Finally, we briefly remark on the invariance property of the path-space measure for such a controlled-diffusion process pertaining to the chain of distributed systems.

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arXiv2015

On the hierarchical optimal control of a chain of distributed systems

In this paper, we consider a chain of distributed systems governed by a degenerate parabolic equation, which satisfies a weak Hörmander type condition, with a control distributed over an open subdomain. In particular, we consider two objectives that we would like to accomplish. The first one being of a controllability type that consists of guaranteeing the terminal state to reach a target set starting from an initial condition; while the second one is keeping the state trajectory of the overall system close to a given reference trajectory on a finite, compact time intervals. We introduce the following framework. First, we partition the control subdomain into two disjoint open subdomains that are compatible with the strategy subspaces of the {\it leader} and that of the {\it follower}, respectively. Then, using the notion of Stackelberg's optimization (which is a hierarchical optimization framework), we provide a new result on the existence of optimal strategies for such an optimization problem -- where the {\it follower} (which corresponds to the second criterion) is required to respond optimally, in the sense of {\it best-response correspondence} to the strategy of the {\it leader} (that is associated to the controllability-type criterion) so as to achieve the overall objectives. Finally, we remark on the implication of our result in assessing the influence of the reachable target set on the optimal strategy of the {\it follower} in relation to the direction of {\it leader-follower} and {\it follower-leader} information flows.

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arXiv2015

Large Deviations for the Reliability Assessment of Redundant Multi-Channel Systems

In this paper, we are concerned with the reliability assessment of redundant multi-channel systems having multiple controllers with overlapping functionality -- where all controllers are required to respond optimally to the non-faulty controllers so as to ensure or maintain some system properties. In particular, for such redundant systems with small random perturbation, we study the relationships between the exit probabilities with which the state-trajectories exit from a given bounded open domain and the value functions corresponding to a family of stochastic exit-time control problems on the boundary of the given domain. Moreover, as the random perturbation vanishes, such relationships provide useful information concerning the reliability of the redundant multi-channel systems arising from the large deviations problem in connection with the asymptotic estimates of exit probabilities with respect to some portions of the boundary of the given domain. Finally, we briefly comment on the implication of our results on a co-design technique using multi-objective optimization frameworks for evaluating the performance of the redundant multi-channel systems.

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arXiv2015

On quantification of systemic redundancy in reliable systems

In this paper, we consider the problem of quantifying systemic redundancy in reliable systems having multiple controllers with overlapping functionality. In particular, we consider a multi-channel system with multi-controller configurations -- where controllers are required to respond optimally, in the sense of best-response correspondence, a {\it reliable-by-design} requirement, to non-faulty controllers so as to ensure or maintain some system properties. Here we introduce a mathematical framework, based on the notion of relative entropy of probability measures associated with steady-state solutions of Fokker-Planck equations for a family of stochastically perturbed multi-channel systems, that provides useful information towards a systemic assessment of redundancy in the system.

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arXiv2014

On the risk-sensitive escape control for diffusion processes pertaining to an expanding construction of distributed control systems

In this paper, we consider an expanding construction of a distributed control system, which is obtained by adding a new subsystem one after the other, until all $n$ subsystems, where $n \ge 2$, are included in the distributed control system. It is assumed that a small random perturbation enters only into the first subsystem and is then subsequently transmitted to the other subsystems. Moreover, for any $\ell \in \{2, ..., n\}$, the distributed control system, compatible with the expanding construction, which is obtained from the first $\ell$ subsystems, satisfies an appropriate Hörmander condition. As a result of this, the diffusion process is degenerate, i.e., the backward operator associated with it is a degenerate parabolic equation. Our main interest here is to prevent the diffusion process (that corresponds to a particular subsystem) from leaving a given bounded open domain. In particular, we consider a risk-sensitive version of the mean escape time criterion with respect to each of the subsystems. Using a variational representation, we characterize the risk-sensitive escape control for the diffusion process as the lower and upper values of an associated stochastic differential game. Finally, we comment on the implication of our results, where one is also interested in evaluating the performance of the risk-sensitive escape control, when there is some modeling error in the distributed control system.

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arXiv2014

On the minimum exit rate for a diffusion process pertaining to a chain of distributed control systems with random perturbations

In this paper, we consider the problem of minimizing the exit rate with which a diffusion process pertaining to a chain of distributed control systems, with random perturbations, exits from a given bounded open domain. In particular, we consider a chain of distributed control systems that are formed by $n$ subsystems (with $n \ge 2$), where the random perturbation enters only in the first subsystem and is then subsequently transmitted to the other subsystems. Furthermore, we assume that, for any $\ell \in \{2, \ldots, n\}$, the distributed control systems, which is formed by the first $\ell$ subsystems, satisfies an appropriate Hörmander condition. As a result of this, the diffusion process is degenerate, in the sense that the infinitesimal generator associated with it is a degenerate parabolic equation. Our interest is to establish a connection between the minimum exit rate with which the diffusion process exits from the given domain and the principal eigenvalue for the infinitesimal generator with zero boundary conditions. Such a connection allows us to derive a family of Hamilton-Jacobi-Bellman equations for which we provide a verification theorem that shows the validity of the corresponding optimal control problems. Finally, we provide an estimate on the attainable exit probability of the diffusion process with respect to a set of admissible (optimal) Markov controls for the optimal control problems.

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arXiv2014

Exit Probabilities for a Chain of Distributed Control Systems with Small Random Perturbations

In this paper, we consider a diffusion process pertaining to a chain of distributed control systems with small random perturbation. The distributed control system is formed by n subsystems that satisfy an appropriate Hormander condition, i.e., the second subsystem assumes the random perturbation entered into the first subsystem, the third subsystem assumes the random perturbation entered into the first subsystem then was transmitted to the second subsystem and so on, such that the random perturbation propagates through the entire distributed control system. Note that the random perturbation enters only in one of the subsystems and, hence, the diffusion process is degenerate, in the sense that the backward operator associated with it is a degenerate parabolic equation. Our interest is to estimate the exit probability with which a diffusion process (corresponding to a particular subsystem) exits from a given bounded open domain during a certain time interval. The method for such an estimate basically relies on the interpretation of the exit probability function as a value function for a family of stochastic control problems that are associated with the underlying chain of distributed control systems.

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arXiv2014

On the Problem of Minimum Asymptotic Exit Rate for Stochastically Perturbed Multi-Channel Dynamical Systems

We consider the problem of minimizing the asymptotic exit rate with which the controlled-diffusion process of a stochastically perturbed multi-channel dynamical system exits from a given bounded open domain. In particular, for a class of admissible bounded linear feedback operators, we establish a connection between the asymptotic exit rate with which such a controlled-diffusion process exits from the given domain and the asymptotic behavior (i.e., a probabilistic characterization) of the principal eigenvalue of the infinitesimal generator, which corresponds to the stochastically perturbed dynamical system, with zero boundary conditions on the given domain. Finally, we briefly remark on the implication of our result for evaluating the performance of the associated deterministic multi-channel dynamical system, when such a dynamical system is composed with a set of (sub)-optimal admissible linear feedback operators.

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arXiv2014

On noncooperative $n$-player principal eigenvalue games

We consider a noncooperative $n$-player principal eigenvalue game which is associated with an infinitesimal generator of a stochastically perturbed multi-channel dynamical system -- where, in the course of such a game, each player attempts to minimize the asymptotic rate with which the controlled state trajectory of the system exits from a given bounded open domain. In particular, we show the existence of a Nash-equilibrium point (i.e., an $n$-tuple of equilibrium linear feedback operators) that is distinctly related to a unique maximum closed invariant set of the corresponding deterministic multi-channel dynamical system, when the latter is composed with this $n$-tuple of equilibrium linear feedback operators.

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arXiv2014

On the entropy of equilibrium measures and game-theoretic equilibrium feedback operators in multi-channel dynamical systems

We investigate the connection between the entropy of equilibrium measures and game-theoretic equilibrium feedback operators in a multi-channel dynamical system. Specifically, we show that the existence of an equilibrium measure, which maximizes the free energy (i.e., the sum of the entropy and the integral over a potential), is related to an equilibrium or "maximum entropy" state for the multi-channel dynamical system that is composed with a set of feedback operators. Further, we observe that such a connection makes sense when this set of feedback operators strategically interacts over an infinite-horizon, in a game-theoretic sense, using the current state-information of the system. Finally, we briefly comment on the implication of our result to the resilient behavior of the equilibrium feedback operators, when there is a random perturbation in the system.

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arXiv2014

Relating maximum entropy, resilient behavior and game-theoretic equilibrium feedback operators in multi-channel systems

In this paper, we first draw a connection between the existence of a stationary density function (which corresponds to an equilibrium state in the sense of statistical mechanics) and a set of feedback operators in a multi-channel system that strategically interacts in a game-theoretic framework. In particular, we show that there exists a set of (game-theoretic) equilibrium feedback operators such that the composition of the multi-channel system with this set of equilibrium feedback operators, when described by density functions, will evolve towards an equilibrium state in such a way that the entropy of the whole system is maximized. As a result of this, we are led to study, by a means of a stationary density function (i.e., a common fixed-point) for a family of Frobenius-Perron operators, how the dynamics of the system together with the equilibrium feedback operators determine the evolution of the density functions, and how this information translates into the maximum entropy behavior of the system. Later, we use such results to examine the resilient behavior of this set of equilibrium feedback operators, when there is a small random perturbation in the system.

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arXiv2013

A brief remark on the topological entropy for linear switched systems

In this brief note, we investigate the topological entropy for linear switched systems. Specifically, we use the Levi-Malcev decomposition of Lie-algebra to establish a connection between the basic properties of the topological entropy and the stability of switched linear systems. For such systems, we show that the topological entropy for the evolution operator corresponding to a semi-simple subalgebra is always bounded from above by the negative of the largest real part of the eigenvalue that corresponds to the evolution operator of a maximal solvable ideal part.

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