Optimal control is the core of modern control theory.

The main problem of its research is:

In order to achieve the optimal control strategy, the performance index is sought to reach its maximum or minimum value while satisfying certain constraints.

**Introduction**

**Introduction**

The basic principles and integrated methods for optimizing performance indicators in control systems can be summarized as follows:

The goal is to find the optimal control strategy from a set of allowed control strategies for a controlled dynamic system or motion process, so that the system’s motion moves from an initial state to a target state, and the performance indicator value is optimized.

These types of problems are prevalent in technical and social issues. For example, determining the optimal control mode to minimize fuel consumption during a spacecraft’s transition from one track to another.

The optimal control theory was formed and developed in the mid-1950s, largely due to advances in space technology. The American scholar R. Berman proposed dynamic programming in 1957 and the principle of maximality, proposed by the former Soviet scholar L.S. Pontiac King in 1958, were key contributions to the development of optimal control theory.

The optimal control problem for linear systems under quadratic performance indicators was solved by R.E. Kalman in the early 1960s.

**Mathematical angle**

**Mathematical angle**

From a mathematical perspective, the optimal control problem can be expressed as follows:

Under the constraints of the motion equation and the allowed control range, the extreme value (maximum or minimum) of the performance index function (referred to as the functional function) is determined, with the control function and motion state as variables.

The main methods for solving the optimal control problem include the classical variational method, the principle of maxima, and dynamic programming.

Optimal control has been applied in various fields such as the design and synthesis of the fastest speed control system, the most fuel-efficient control system, the system with minimum energy consumption, linear regulators, and more.

Variational theory is a powerful mathematical tool for studying optimal control problems. Classical variational theory can only solve problems without constraints, but most control problems in engineering practice involve constraints, leading to the development of modern variational theory.

**Research method**

**Research method**

The two most commonly used methods in modern variational theory are the dynamic programming method and the principle of minimum value. Both methods are capable of solving variational problems with closed set constraints.

It is important to note that the dynamic programming method and the principle of minimum value are essentially analytical methods.

Other analytical methods for solving the optimal control problem include the variational method and the linear quadratic control method.

In addition to analytical methods, research on the optimal control problem also includes numerical methods and gradient methods.