Optimal control theory is a mathematical optimization technique used in science and engineering to determine control policies. It uses various methods of analysis to determine the parameters of a system and is useful for solving continuous-time optimization problems. The theory can be derived using Pontryagin’s maximum principle or the Hamilton-Jacobi-Bellman equation. Examples of its use include optimizing travel time and fuel consumption while accounting for constraints. It can also be used for cost or shadow price resolution.
Optimal control theory is used extensively in science and engineering. It is a mathematical optimization technique commonly used in the creation of control policies. Lev Pontryagin, together with his team in the former Soviet Union, and the American Richard Bellman are primarily responsible for the theory of optimal control. The general purpose of the theory is to use various methods of analysis to determine the parameters of a system by conducting trial and error processes.
Optimal control theory is useful when trying to solve continuous-time optimization problems. The theory addresses a problem by determining a control law for a hypothetical system in order to achieve a level of optimality. The optimal control consists of a set of various equations, which describe the paths of the variables that bring the cost functional to a minimum. The cost functional is basically a function of state and control variables. Optimal control theory makes use of Pontryagin’s maximum principle, which generally states that one can solve the optimization problem P with the use of a Hamiltonian function H over a period, which is a necessary condition. The theory can also be derived with the Hamilton-Jacobi-Bellman equation.
To help a person understand optimal control theory, the example “driving the car on a hilly road” is commonly used. Imagine traveling in an automobile up a steep road in a straight line. Theory can determine how one should accelerate to minimize absolute travel time. In this case the “system” consists of the vehicle and the stony road and the criterion of optimality is precisely that of minimizing the travel time. Such problems are known to include constraints (e.g. fuel caps, speed limits). Another question could be to find a way to optimize the fuel consumption of the car while being forced to complete a certain route in a certain time limit.
Another example of the use of optimal control theory is cost or shadow price resolution. It consists of the marginal value of the expansion of the state variable. Once this is resolved, the optimal value for the control can form a differential equation conditional on the awareness of the constant. It is common for this strategy to resolve for regions describing optimal control and isolate the actual choice values over time.
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