Mathematical Control Theory

According to E. D. Sontag, Mathematical Control Theory “is the area of application-oriented mathematics that deals with the basic principles underlying the analysis and design of control systems”. In other words, mathematical control theory studies the properties of control systems. Roughly speaking a control system is a more general notion than the notion of a dynamical system and is an “object” that changes with time and its behavior can be influenced by external parameters (inputs). Control systems are divided into two groups, the stochastic control systems (systems for which the future state is not completely determined by the present and the external parameters) and the deterministic control systems (systems for which the future state is completely determined by the present state and the external parameters). 

The mathematician N. Wiener coined the term “Cybernetics” to refer to control theory and related areas.

Directions of Mathematical Control Theory

There are three directions in Mathematical Control Theory:

1) The direction of Optimal Control: Studies the ability to optimize the behavior of a control system 

2) The direction of Controllability and Observability Problems: Studies the controllability and observability properties of a control system

3) The direction of Feedback Control: Studies the ability to stabilize a control system by means of feedback control

Roughly speaking,

- in the direction of optimal control the control action (input) is pre-computed and then it is applied to the system in order to obtain the optimal behavior. The resulting system is an open-loop system.

- in the direction of controllability and observability the problem of the existence of a control action (input) that can achieve certain objectives is studied (or the problem of estimating the state of a system based on limited information).

- in the direction of feedback control the control action (input) is calculated and applied on-line (i.e., as the system evolves) in order to obtain the desired behavior. Specifically, some variables of the system (the output) are measured and the values of the measured variables determine the control action at each time through a law, which is termed as the “feedback law” (because schematically the measured variables are “fed back” to the system). The resulting system is a closed-loop system.

The Direction of Feedback Control

I am most interested in the third direction of Mathematical Control Theory (feedback control), although as it is pointed out in many texts all directions of Mathematical Control Theory are complementary. 

It can be said that the direction of feedback control studies two major problems: the problem of existence of a stabilizing feedback for a given control system and the problem of the design of stabilizing feedback for a given control system.  Many classical control problems can be recast as feedback control problems. Particularly, this holds for the tracking control problem and the observer problem (existence and design). Moreover, the mathematical framework used in the direction of feedback control is adequate for the formulation of feedback control problems for:

- discrete-time systems (finite or infinite dimensional)

- systems described by ordinary differential equations (finite-dimensional continuous-time systems)

- certain hybrid systems

- systems described by delay differential equations

- systems described by partial differential equations

The direction of Feedback Control in Mathematical Control Theory deals with the theory and mathematics of the area of Automatic Control (a very important area in engineering). In fact, most of the technological achievements of the 20th and 21st century contain a control mechanism, which was designed using mathematical results of the direction of Feedback Control in Mathematical Control Theory. The direction of Feedback Control in Mathematical Control Theory is also related with the applied areas of Signal Processing and Systems Theory.

Areas of Mathematics related to Feedback control

It should be clear that the direction of feedback control is closely related to

A) the Stability Theory (or Qualitative Theory) of Dynamical Systems.

B) the area of (partial or ordinary) Differential and Difference Equations (existence and uniqueness theory)

C) the areas of Non-Smooth Analysis and Set-Valued Analysis. 

A student interested in the direction of feedback control should have a solid background on advanced calculus, real analysis, linear algebra and differential equations.

Mathematical Control Theory’s Journals

1) SIAM Journal on Control and Optimization -> here

2) Mathematics of Control, Signals and Systems -> here

3) International Journal of Control -> here

4) IMA Journal of Mathematical Control and Information -> here

5) Systems and Control Letters -> here

6) IEEE Transactions on Automatic Control -> here

7) International Journal of Robust and Nonlinear Control -> here

8) Automatica -> here

9) Journal of the Franklin Institute -> here

10) European Journal of Control -> here

11) ESAIM Control, Optimisation and Calculus of Variations -> here

There are many more journals publishing engineering-oriented results, which are not listed above. 

Moreover, there are journals focused on areas of mathematics related to mathematical control theory where one can find important papers on control issues (sometimes viewed from a wider or narrower perspective).