# What is Optimal Control

##### Optimal Control: An Introduction

Optimal control is a branch of control theory that deals with finding the control inputs for a dynamical system to achieve a specific goal. The goal could be to minimize the cost of some operation, maximize the performance of a system, or ensure that the system operates within certain constraints. Optimal control is used in many fields such as engineering, economics, biology, and physics to control and optimize complex systems.

In this article, we will introduce the concept of optimal control and discuss different approaches to solving optimal control problems. We will also present some real-world examples to demonstrate the practical applications of optimal control.

##### Mathematical Formulation of Optimal Control

In optimal control, we aim to find the control inputs u(t) that minimize a cost function J over a given time interval [t1, t2]. The cost function is typically defined as the sum of a running cost L and a terminal cost M:

• Running Cost: The running cost L is a function of the state x(t) and control input u(t) and is integrated over the time interval [t1, t2].
• Terminal Cost: The terminal cost M is a function of the state x(t2) at the final time t2.

The goal is to find the control inputs u(t) that minimize the cost function J:

J[u] = ∫t1t2 L(x(t), u(t), t)dt + M(x(t2))

Such a problem is known as an optimal control problem. It is subject to a set of differential equations known as the state equations, which describe the dynamics of the system:

ẋ(t) = f(x(t), u(t), t)

Here, x(t) is the state of the system at time t, ẋ(t) is the derivative of the state with respect to time, u(t) is the control input, and f is a nonlinear function that describes the dynamics of the system. The state equations are used to describe the evolution of the system over time.

The optimal control problem is subject to some constraints, such as boundary conditions on the states and controls, and constraints on the allowable control inputs. These constraints can be incorporated into the optimization problem using the method of Lagrange multipliers, resulting in the following optimality conditions:

• State Equation: ẋ*(t) = f*(x(t), u*(t), t);
• Adjoint Equation: λ*(t) = -∂H/∂x, λ(t1) = ∂M/∂x(t1), where H is the Hamiltonian of the system;
• Optimality Condition: u*(t) = argminu{H(x*(t), u, λ*(t), t)}.

These equations represent the necessary conditions for an optimal control problem. They form the basis of various numerical methods used to solve optimal control problems.

##### Approaches to Solve Optimal Control Problems

There are different approaches to solve optimal control problems, depending on the complexity of the system and the requirements of the problem. We will discuss the following approaches:

• Pontryagin's Maximum Principle: This is a method for finding necessary conditions for optimality. It applies the methodology of calculus of variations to optimal control problems. The maximum principle states that the optimal control u*(t) minimizes the Hamiltonian at every point in time. This principle enables us to derive the necessary conditions for optimality, as mentioned above.
• Dynamic Programming: Dynamic programming is an approach used to solve optimal control problems with discrete time and finite horizon. The method involves solving sub-problems and using the solutions to solve the overall problem. This approach requires the system to have the Markov property, which means that the future state of the system depends only on its current state and the current control input.
• Linear Quadratic Regulator: The linear quadratic regulator (LQR) is an approach used to solve optimal control problems for linear time-invariant systems with quadratic cost functions. The method involves solving the Riccati differential equation to obtain the optimal feedback control. The LQR approach is widely used in control engineering to design optimal controllers for linear systems.
• Model Predictive Control: Model predictive control (MPC) is an approach used to solve optimal control problems for systems with complicated dynamics and constraints. The method involves solving a finite-horizon optimal control problem at each time step and using the predicted future states to compute the optimal control sequence. The MPC approach is widely used in process control, robotics, and autonomous systems.
##### Real-World Applications of Optimal Control

Optimal control has many real-world applications in different fields such as aerospace engineering, robotics, economics, and healthcare. Here, we present some examples of how optimal control is used in solving practical problems.

• Aerospace Engineering: Optimal control is used in aerospace engineering to optimize the trajectory of spacecraft, missiles, and satellites. For example, in the Apollo mission, optimal control was used to design the trajectory of the spacecraft to reach the moon and return to earth safely. Similarly, optimal control is used to design the trajectory of missiles to hit a moving target accurately.
• Robotics: Optimal control is used in robotics to design controllers that enable robots to perform complex tasks. For instance, in the motion planning of a robot arm, optimal control is used to determine the path of the end effector that accomplishes the task with minimum energy consumption. Similarly, in the locomotion of a humanoid robot, optimal control is used to design the gait that minimizes the energy consumption and maximizes the stability of the robot.
• Economics: Optimal control is used in economics to model and predict the behavior of the economy. For example, optimal control is used in macroeconomic policy to optimize the inflation rate, unemployment rate, and output level of a country. Similarly, optimal control is used in financial engineering to design portfolios that maximize the return on investment and minimize the risk.
• Healthcare: Optimal control is used in healthcare to design personalized treatment plans for patients. For instance, in the management of diabetes, optimal control is used to design the insulin dosage that maintains the blood glucose level within the desired range. Similarly, in the treatment of cancer, optimal control is used to design the chemotherapy dosage and schedule that maximizes the tumor kill rate and minimizes the side-effects.
##### Conclusion

Optimal control is a powerful tool for designing controllers that optimize the performance of complex systems. It involves finding the control inputs that minimize a cost function over a given time interval, subject to dynamic and control constraints. Various approaches exist to solve optimal control problems, depending on the nature of the system and the requirements of the problem. Optimal control has many real-world applications, ranging from aerospace engineering to healthcare. Solving optimal control problems requires a combination of mathematical and computational tools, making optimal control a multidisciplinary field.