Agamirza E. Bashirov

On Concepts of Controllability for Deterministic and Stochastic Systems

The new necessary and sufficient conditions, which are formulated in terms of convergence of a certain sequence of operators involving the resolvent of the negative of the controllability operator, are found for deterministic linear stationary control systems to be completely and approximately controllable, respectively. These conditions are applied to study the S-controllability (a property of attaining an arbitrarily small neighborhood of each point in the state space with a probability arbitrarily close to one) and C--controllability (the S--controllability fortified with some uniformity) of stochastic systems. It is shown that the S--controllability (the C--controllability) of a partially observable linear stationary control system with an additive Gaussian white noise disturbance on all the intervals [0,T] for T>0 is equivalent to the approximate (complete) controllability of its deterministic part on all the intervals [0,T] for T>0.

On Controllability Conception for Stochastic Systems

The controllability notions for partially observed stochastic systems are defined. Their relation with complete and approximate controllabilities is shown. In particular, it is proven that the approximate controllability condition is necessary and the complete controllability condition is sufficient for the partially observed linear Gaussian control system to attain the arbitrarily small neighborhood of each point in the state space with probability arbitrarily closely to one.

Partial controllability concepts

The main results in theory of controllability are formulated for deterministic or stochastic control systems given in a standard form. i.e., given as a first order differential equation driven by an infinitesimal generator of strongly continuous semigroup in an abstract Hilbert or Banach space and disturbed by a deterministic function or by a white noise process. At the same time, some deterministic or stochastic linear systems can be written in a standard form if the state space is enlarged. Respectively, the ordinary controllability conditions for them are too strong since they assume extended state space. It is reasonable to introduce partial controllability concepts, which assume original state space. In this paper, we study necessary and sufficient conditions of partial controllability for deterministic and stochastic linear control systems given in a standard form and their implications to particular cases.

Partially Observable Linear Systems Under Dependent Noises

1 Basic Elements of Functional Analysis.- 1.1 Sets and Functions.- 1.1.1 Sets and Quotient Sets.- 1.1.2 Systems of Numbers and Cardinality.- 1.1.3 Systems of Sets.- 1.1.4 Functions and Sequences.- 1.2 Abstract Spaces.- 1.2.1 Linear Spaces.- 1.2.2 Metric Spaces.- 1.2.3 Banach Spaces.- 1.2.4 Hilbert and Euclidean Spaces.- 1.2.5 Measurable and Borel Spaces.- 1.2.6 Measure and Probability Spaces.- 1.2.7 Product of Spaces.- 1.3 Linear Operators.- 1.3.1 Bounded Operators.- 1.3.2 Inverse Operators.- 1.3.3 Closed Operators.- 1.3.4 Adjoint Operators.- 1.3.5 Projection Operators.- 1.3.6 Self-Adjoint, Nonnegative and Coercive Operators.- 1.3.7 Compact, Hilbert-Schmidt and Nuclear Operators.- 1.4 Weak Convergence.- 1.4.1 Strong and Weak Forms of Convergence.- 1.4.2 Weak Convergence and Convexity.- 1.4.3 Convergence of Operators.- 2 Basic Concepts of Analysis in Abstract Spaces.- 2.1 Continuity.- 2.1.1 Continuity of Vector-Valued Functions.- 2.1.2 Weak Lower Semicontinuity.- 2.1.3 Continuity of Operator-Valued Functions.- 2.2 Differentiability.- 2.2.1 Differentiability of Nonlinear Operators.- 2.2.2 Differentiability of Operator-Valued Functions.- 2.3 Measurability.- 2.3.1 Measurability of Vector-Valued Functions.- 2.3.2 Measurability of Operator-Valued Functions.- 2.3.3 Measurability of G1- and G2-Valued Functions.- 2.4 Integrability.- 2.4.1 Bochner Integral.- 2.4.2 Fubinis Property.- 2.4.3 Change of Variable.- 2.4.4 Strong Bochner Integral.- 2.4.5 Bochner Integral of G1- and G2-Valued Functions.- 2.5 Integral and Differential Operators.- 2.5.1 Integral Operators.- 2.5.2 Integral Hilbert-Schmidt Operators.- 2.5.3 Differential Operators.- 2.5.4 Gronwalls Inequality and Contraction Mappings.- 3 Evolution Operators.- 3.1 Main Classes of Evolution Operators.- 3.1.1 Strongly Continuous Semigroups.- 3.1.2 Examples.- 3.1.3 Mild Evolution Operators.- 3.2 Transformations of Evolution Operators.- 3.2.1 Bounded Perturbations.- 3.2.2 Some Other Transformations.- 3.3 Operator Riccati Equations.- 3.3.1 Existence and Uniqueness of Solution.- 3.3.2 Dual Riccati Equation.- 3.3.3 Riccati Equations in Differential Form.- 3.4 Unbounded Perturbation.- 3.4.1 Preliminaries.- 3.4.2 ?*-Perturbation.- 3.4.3?-Perturbation.- 3.4.4 Examples.- 4 Partially Observable Linear Systems.- 4.1 Random Variables and Processes.- 4.1.1 Random Variables.- 4.1.2 Conditional Expectation and Independence.- 4.1.3 Gaussian Systems.- 4.1.4 Random Processes.- 4.2 Stochastic Modelling of Real Processes.- 4.2.1 Brownian Motion.- 4.2.2 Wiener Process Model of Brownian Motion.- 4.2.3 Diffusion Processes.- 4.3 Stochastic Integration in Hilbert Spaces.- 4.3.1 Stochastic Integral.- 4.3.2 Martingale Property.- 4.3.3 Fubinis Property.- 4.3.4 Stochastic Integration with Respect to Wiener Processes.- 4.4 Partially Observable Linear Systems.- 4.4.1 Solution Concepts.- 4.4.2 Linear Stochastic Evolution Systems.- 4.4.3 Partially Observable Linear Systems.- 4.5 Basic Estimation in Hilbert Spaces.- 4.5.1 Estimation of Random Variables.- 4.5.2 Estimation of Random Processes.- 4.6 Improving the Brownian Motion Model.- 4.6.1 White, Colored and Wide Band Noise Processes.- 4.6.2 Integral Representation of Wide Band Noises.- 5 Separation Principle.- 5.1 Setting of Control Problem.- 5.1.1 State-Observation System.- 5.1.2 Set of Admissible Controls.- 5.1.3 Quadratic Cost Functional.- 5.2 Separation Principle.- 5.2.1 Properties of Admissible Controls.- 5.2.2 Extended Separation Principle.- 5.2.3 Classical Separation Principle.- 5.2.4 Proof of Lemma 5.15.- 5.3 Generalization to a Game Problem.- 5.3.1 Setting of Game Problem.- 5.3.2 Case 1: The First Player Has Worse Observations.- 5.3.3 Case 2: The Players Have the Same Observations.- 5.4 Minimizing Sequence.- 5.4.1 Properties of Cost Functional.- 5.4.2 Minimizing Sequence.- 5.5 Linear Regulator Problem.- 5.5.1 Setting of Linear Regulator Problem.- 5.5.2 Optimal Regulator.- 5.6 Existence of Optimal Control.- 5.6.1 Controls in Linear Feedback Form.- 5.6.2 Existence of Optimal Control.- 5.6.3 Application to Existence of Saddle Points.- 5.7 Concluding Remarks.- 6 ntrol and Estimation under Correlated White Noises.- 6.1 Estimation: Preliminaries.- 6.1.1 Setting of Estimation Problems.- 6.1.2 Wiener-s Property.- 4.3.4 Stochastic Integration with Respect to Wiener Processes.- 4.4 Partially Observable Linear Systems.- 4.4.1 Solution Concepts.- 4.4.2 Linear Stochastic Evolution Systems.- 4.4.3 Partially Observable Linear Systems.- 4.5 Basic Estimation in Hilbert Spaces.- 4.5.1 Estimation of Random Variables.- 4.5.2 Estimation of Random Processes.- 4.6 Improving the Brownian Motion Model.- 4.6.1 White, Colored and Wide Band Noise Processes.- 4.6.2 Integral Representation of Wide Band Noises.- 5 Separation Principle.- 5.1 Setting of Control Problem.- 5.1.1 State-Observation System.- 5.1.2 Set of Admissible Controls.- 5.1.3 Quadratic Cost Functional.- 5.2 Separation Principle.- 5.2.1 Properties of Admissible Controls.- 5.2.2 Extended Separation Principle.- 5.2.3 Classical Separation Principle.- 5.2.4 Proof of Lemma 5.15.- 5.3 Generalization to a Game Problem.- 5.3.1 Setting of Game Problem.- 5.3.2 Case 1: The First Player Has Worse Observations.- 5.3.3 Case 2: The Players Have the Same Observations.- 5.4 Minimizing Sequence.- 5.4.1 Properties of Cost Functional.- 5.4.2 Minimizing Sequence.- 5.5 Linear Regulator Problem.- 5.5.1 Setting of Linear Regulator Problem.- 5.5.2 Optimal Regulator.- 5.6 Existence of Optimal Control.- 5.6.1 Controls in Linear Feedback Form.- 5.6.2 Existence of Optimal Control.- 5.6.3 Application to Existence of Saddle Points.- 5.7 Concluding Remarks.- 6 ntrol and Estimation under Correlated White Noises.- 6.1 Estimation: Preliminaries.- 6.1.1 Setting of Estimation Problems.- 6.1.2 Wiener-Hopf Equation.- 6.2 Filtering.- 6.2.1 Dual Linear Regulator Problem.- 6.2.2 Optimal Linear Feedback Filter.- 6.2.3 Error Process.- 6.2.4 Innovation Process.- 6.3 Prediction.- 6.3.1 Dual Linear Regulator Problem.- 6.3.2 Optimal Linear Feedback Predictor.- 6.4 Smoothing.- 6.4.1 Dual Linear Regulator Problem.- 6.4.2 Optimal Linear Feedback Smoother.- 6.5 Stochastic Regulator Problem.- 6.5.1 Setting of the Problem.- 6.5.2 Optimal Stochastic Regulator.- 7 Control and Estimation under Colored Noises.- 7.1 Estimation.- 7.1.1 Setting of Estimation Problems.- 7.1.2 Reduction.- 7.1.3 Optimal Linear Feedback Estimators.- 7.1.4 About the Riccati Equation (7.15).- 7.1.5 Example: Optimal Filter in Differential Form.- 7.2 Stochastic Regulator Problem.- 7.2.1 Setting of the Problem.- 7.2.2 Reduction.- 7.2.3 Optimal Stochastic Regulator.- 7.2.4 About the Riccati Equation (7.48).- 7.2.5 Example: Optimal Stochastic Regulator in Differential Form.- 8 Control and Estimation under Wide Band Noises.- 8.1 Estimation.- 8.1.1 Setting of Estimation Problems.- 8.1.2 The First Reduction.- 8.1.3 The Second Reduction.- 8.1.4 Optimal Linear Feedback Estimators.- 8.1.5 About the Riccati Equation (8.40).- 8.1.6 Example: Optimal Filter in Differential Form.- 8.2 More About the Optimal Filter.- 8.2.1 More About the Riccati Equation (8.40).- 8.2.2 Equations for the Optimal Filter.- 8.3 Stochastic Regulator Problem.- 8.3.1 Setting of the Problem.- 8.3.2 Reduction.- 8.3.3 Optimal Stochastic Regulator.- 8.3.4 About the Riccati Equation (8.81).- 8.3.5 Example: Optimal Stochastic Regulator in Differential Form.- 8.4 Concluding Remarks.- 9 Control and Estimation under Shifted White Noises.- 9.1 Preliminaries.- 9.2 State Noise Delaying Observation Noise: Filtering.- 9.2.1 Setting of the Problem.- 9.2.2 Dual Linear Regulator Problem.- 9.2.3 Optimal Linear Feedback Filter.- 9.2.4 About the Riccati Equation (9.27).- 9.2.5 About the Optimal Filter.- 9.3 State Noise Delaying Observation Noise: Prediction.- 9.4 State Noise Delaying Observation Noise: Smoothing.- 9.5 State Noise Delaying Observation Noise: Stochastic Regulator Prob-lem.- 9.6 Concluding Remarks.- 10 Control and Estimation under Shifted White Noises (Revised).- 10.1 Preliminaries.- 10.2 Shifted White Noises and Boundary Noises.- 10.3 Convergence of Wide Band Noise Processes.- 10.3.1 Approximation of White Noises.- 10.3.2 Approximation of Shifted White Noises.- 10.4 State Noise Delaying Observation Noise.- 10.4.1 Setting of the Problem.- 10.4.2 Approximating Problems.- 10.4.3 Optimal Control and Optimal Filter.- 10.4.4 Application to Space Navigation and Guidance.- 10.5 State Noise Anticipating Observation Noise.- 10.5.1 Setting of the Problem.- 10.5.2 Approximating Problems.- 10.5.3 Optimal Control and Optimal Filter.- 11 Duality.- 11.1 Classical Separation Principle and Duality.- 11.2 Extended Separation Principle and Duality.- 11.3 Innovation Process for Control Actions.- 12 Controllability.- 12.1 Preliminaries.- 12.1.1 Definitions.- 12.1.2 Description of the System.- 12.2 Controllability: Deterministic Systems.- 12.2.1 CCC, ACC and Rank Condition.- 12.2.2 Resolvent Conditions.- 12.2.3 Applications of Resolvent Conditions.- 12.3 Controllability: Stochastic Systems.- 12.3.1 ST-Controllability.- 12.3.2 CT-Controllability.- 12.3.3 ST-Controllability.- Comments.- Bibiography.- Index of Notation.

On weakening of the controllability concepts

The controllability concept for the partially observed stochastic system is defined as the property of attaining the arbitrarily small neighborhood of each point in the state space with probability arbitrarily closely to one. It is shown that the approximate controllability condition is necessary and the complete controllability condition is sufficient for the partially observed linear Gaussian control system to be controllable in the above sense.

Partial controllability of stochastic linear systems

The controllability concepts for linear stochastic differential equations, driven by different kinds of noise processes, can be reduced to the partial controllability concepts for the same systems, driven by correlated white noises. Based on this fact, in this article, we study the conditions of exact and approximate controllability for linear stochastic control systems under various kinds of noise processes, including correlated white noises as well as coloured, wide band and shifted white noises. It is proved that such systems are never exactly controllable while their approximate controllability is equivalent to the approximate controllability of the associated linear deterministic systems at all past time moments.

Controllability of linear deterministic and stochastic systems

Necessary and sufficient conditions, which are formulated in terms of convergence of a certain sequence of operators involving the resolvent of the negative of the controllability operator, are found for deterministic linear stationary control systems to be completely and approximately controllable, respectively. It is shown that the S-controllability of partially observable linear stationary control systems with additive Gaussian white noise disturbance, that is a property of attaining an arbitrarily small neighborhood of each point in the state space with a probability arbitrarily close to one, is equivalent to the approximate controllability of their deterministic part. Also, it is shown that the C-controllability of the same kind of systems, that is defined as S-controllability improved with some uniformity conditions, is equivalent to the complete controllability of their deterministic part.

On multiplicative and Volterra minimization methods

Theory and applications of multiplicative and Volterra calculi have been evolving rapidly over the recent years. As numerical minimization methods have a wide range of applications in science and engineering, the idea of the design of minimization methods based on multiplicative and Volterra calculi is self-evident. In this paper, the well-known Newton minimization method for one and two variables is developed in the frameworks of multiplicative and Volterra calculi. The efficiency of these proposed minimization methods is exposed by examples, and the results are compared with the original minimization method. One of the striking results of the proposed method is that the rate of convergence and the range of initial values are considerably larger compared to the original method.

On partial approximate controllability of semilinear systems

Abstract In this paper, a sufficient condition for the partial approximate controllability of semilinear deterministic control systems is proved. Generally, the theorems on controllability are formulated for control systems given as a first-order differential equation, while many systems can be written in this form only by enlarging the dimension of the state space. The ordinary controllability conditions for such systems are too strong because they involve the enlarged state space. Therefore, it becomes useful to define partial controllability concepts, which assume the original state space. The method of proof, given in this paper, differs from the traditional proofs by fixed point theorems. The obtained result is demonstrated on examples.

On Partial Complete Controllability of Semilinear Systems

Many control systems can be written as a first-order differential equation if the state space enlarged. Therefore, general conditions on controllability, stated for the first-order differential equations, are too strong for these systems. For such systems partial controllability concepts, which assume the original state space, are more suitable. In this paper, a sufficient condition for the partial complete controllability of semilinear control system is proved. The result is demonstrated through examples.