## Final Activity Report Summary - ICCCSYSTEMS (Robust Information Transmission and Control Subject to uncertainty, capacity and power constraints)

During the first 12 month the following work has been completed,

1) Robust Control of Uncertain Stochastic Systems;

2) Robust Information Transmission for Uncertain Systems.

In addition, part of the work associated with 3) Relations between Robustness of Control Systems and Robustness of Information Transmission is completed.

The results obtained corresponding to the above activities 1)-3) are the following.

1) Formulation and solution of the robust control for uncertain systems, existence, uniqueness and the optimal solution. The results are either published or submitted for publication.

2) Formulation and solution of robust source coding and robust capacity. The results include, new lossless uniquely decodable codes, new capacity formulas when the channel is static as well as dynamic. The results are either published or submitted for publication.

3) Formulation and identification of various relations between robust control techniques (games, induced norms, dissipation) and information theoretic techniques (entropy, relative entropy). The results are either published or submitted for publication.

During the last 12 month the following work has been completed,

1) Relations between robustness of control systems and robustness of information theory;

2) Convergence of communication and control subject to capacity and power constraints.

The results obtained corresponding to the above activities 1)-2) are the following.

1) Formulation and solution of the robust control for uncertain systems, existence, uniqueness and the optimal solution. Formulation and solution of robust capacity and relations to robustness of control systems. This part consider minimax games for stochastic uncertain systems with the pay-off being a nonlinear functional of the uncertain measure where the uncertainty is measured in terms of relative entropy between the uncertain and the nominal measure. The maximizing player is the uncertain measure, while the minimizer is the control which induces a nominal measure. Existence and uniqueness of minimax solutions are derived on suitable spaces of measures. Several examples are presented illustrating the results. Subsequently, the results are also applied to controlled Stochastic Differential Equations (SDE's) on Hilbert Spaces. Based on infinite dimensional extension of Girsanov's measure transformation, martingale solutions are used in establishing existence and uniqueness of minimax strategies. Moreover, some basic properties of the relative entropy of measures on infinite dimensional spaces are presented. Similar formulations are also introduced for computing the channel capacity when the channel is unknown but described by relative entropy constraints. Much of the work applies to Information theory problems as well.

2) Formulation and solution of control over limited capacity communication channels, by finding encoders, decoders, controllers which achieve stability and observability. Moreover, for linear Gaussian control systems over Additive Noise channels, a separation is establish for designing the communication system (encoder. decoder) and the control system (state estimator and controller) .

In this part of the work, first we establish a causal information theory which is applicable to control systems. Then we show that an innovations encoder is the appropriate encoder when dealing with partially observed systems. Subsequently, we design an encoder decoder so that the source is matched to the channel. Finally, we design the controller to achieve stabilisation. Our results are applicable to nonlinear control systems as well.

1) Robust Control of Uncertain Stochastic Systems;

2) Robust Information Transmission for Uncertain Systems.

In addition, part of the work associated with 3) Relations between Robustness of Control Systems and Robustness of Information Transmission is completed.

The results obtained corresponding to the above activities 1)-3) are the following.

1) Formulation and solution of the robust control for uncertain systems, existence, uniqueness and the optimal solution. The results are either published or submitted for publication.

2) Formulation and solution of robust source coding and robust capacity. The results include, new lossless uniquely decodable codes, new capacity formulas when the channel is static as well as dynamic. The results are either published or submitted for publication.

3) Formulation and identification of various relations between robust control techniques (games, induced norms, dissipation) and information theoretic techniques (entropy, relative entropy). The results are either published or submitted for publication.

During the last 12 month the following work has been completed,

1) Relations between robustness of control systems and robustness of information theory;

2) Convergence of communication and control subject to capacity and power constraints.

The results obtained corresponding to the above activities 1)-2) are the following.

1) Formulation and solution of the robust control for uncertain systems, existence, uniqueness and the optimal solution. Formulation and solution of robust capacity and relations to robustness of control systems. This part consider minimax games for stochastic uncertain systems with the pay-off being a nonlinear functional of the uncertain measure where the uncertainty is measured in terms of relative entropy between the uncertain and the nominal measure. The maximizing player is the uncertain measure, while the minimizer is the control which induces a nominal measure. Existence and uniqueness of minimax solutions are derived on suitable spaces of measures. Several examples are presented illustrating the results. Subsequently, the results are also applied to controlled Stochastic Differential Equations (SDE's) on Hilbert Spaces. Based on infinite dimensional extension of Girsanov's measure transformation, martingale solutions are used in establishing existence and uniqueness of minimax strategies. Moreover, some basic properties of the relative entropy of measures on infinite dimensional spaces are presented. Similar formulations are also introduced for computing the channel capacity when the channel is unknown but described by relative entropy constraints. Much of the work applies to Information theory problems as well.

2) Formulation and solution of control over limited capacity communication channels, by finding encoders, decoders, controllers which achieve stability and observability. Moreover, for linear Gaussian control systems over Additive Noise channels, a separation is establish for designing the communication system (encoder. decoder) and the control system (state estimator and controller) .

In this part of the work, first we establish a causal information theory which is applicable to control systems. Then we show that an innovations encoder is the appropriate encoder when dealing with partially observed systems. Subsequently, we design an encoder decoder so that the source is matched to the channel. Finally, we design the controller to achieve stabilisation. Our results are applicable to nonlinear control systems as well.