Question

Find the optimal paths of the control, state, and costate variables that will $$\begin{array}{ll} \text { Maximize } & \int_{0}^{T}\left(y u-u^{2}-y^{2}\right) d t \\ \text { subject to } & y^{\prime}=u \\ \text { and } & y(0)=y_{0} \quad y(t) \text { free } \end{array}$$

Short answer

Maximize by setting up Hamiltonian, solving Euler-Lagrange equations, and applying boundary conditions.

Step-by-step solution

Setup Hamiltonian

First, we need to set up the Hamiltonian for the given problem. The Hamiltonian, denoted by \( H \), is constructed from the integrand of the objective function and the constraint using a Lagrange multiplier (costate) \( \lambda(t) \). Here, the Hamiltonian is given by:\[ H = yu - u^2 - y^2 + \lambda(t)u \]

Write Euler-Lagrange Equations

The Euler-Lagrange equations for the control \( u \), state \( y \), and costate \( \lambda(t) \) are derived from the Hamiltonian. They consist of the following set of equations:- The control equation, by maximizing \( H \) with respect to \( u \): \[ \frac{\partial H}{\partial u} = y - 2u + \lambda = 0 \]- The state equation, which is simply \( y' = u \).- The costate equation, derived from \( -\frac{d\lambda}{dt} = \frac{\partial H}{\partial y} \): \[ \frac{d\lambda}{dt} = 2y \]

Solve for Optimal Control

From the control equation, solve for the control variable \( u \):\[ y - 2u + \lambda = 0 \Rightarrow u = \frac{y + \lambda}{2} \]

Solve State and Costate Equations

Substitute the expression for \( u \) back into the state equation \( y' = u \) and the costate equation \( \frac{d\lambda}{dt} = 2y \). This gives two first-order differential equations:1. \( y' = \frac{y + \lambda}{2} \)2. \( \frac{d\lambda}{dt} = 2y \)These can be solved simultaneously with appropriate initial conditions.

Apply Boundary Conditions

Use the initial condition \( y(0) = y_0 \) and the transversality condition \( \lambda(T) = 0 \), given \( y(t) \) is free at \( t = T \), to solve for constants of integration and find the explicit paths of \( y(t) \), \( u(t) \), and \( \lambda(t) \).

Key concepts

Hamiltonian Mechanics

Hamiltonian Mechanics is a powerful framework used to describe the evolution of a system in physics and mathematics. It provides a different perspective compared to Newtonian mechanics, focusing on energy conservation and symmetries. In optimal control theory, the Hamiltonian plays a pivotal role in deriving optimal control laws. For the given problem, the Hamiltonian integrates both the objective function and the dynamics of the system using a Lagrange multiplier, often termed as costate variable, denoted here by \( \lambda(t) \). - The Hamiltonian function \( H \) encapsulates the system's constraints and objectives. - For our problem, \[ H = yu - u^2 - y^2 + \lambda(t)u \] balances the contribution of the control \( u \), the state \( y \), and the costate variable \( \lambda(t) \).This function helps us find the control \( u \) that maximizes the trajectory of the system by incorporating both the direct and indirect effects of the control and state variables.

Euler-Lagrange Equations

The Euler-Lagrange equations are cornerstone equations in calculus of variations, used to find functions that optimize a certain functional. In the context of optimal control, they are derived from the Hamiltonian to determine the optimal path for control, state, and costate variables.For the control equation, we maximize the Hamiltonian with respect to the control \( u \): - \[ \frac{\partial H}{\partial u} = y - 2u + \lambda = 0 \]- Solving this gives: \[ u = \frac{y + \lambda}{2} \]For the state equation, we directly use the given differential equation:- \[ y' = u \]Finally, the costate equation is derived as:- \[-\frac{d\lambda}{dt} = \frac{\partial H}{\partial y} = -2y \]- Leading to: \[ \frac{d\lambda}{dt} = 2y \]These interrelated equations must be solved simultaneously to find the optimal solutions.

Differential Equations

Differential equations play a fundamental role in modeling and solving dynamic systems. In this context, they describe how the system evolves over time, dictated by both the state and costate variables.In the given problem, two crucial first-order differential equations arise:- The state equation: \[ y' = \frac{y + \lambda}{2} \]- The costate equation: \[ \frac{d\lambda}{dt} = 2y \]These equations inform us about the rates of change of the state and costate variables and their interdependency. Solving them requires appropriate boundary conditions, leading us to the transversality condition next. It is a step-by-step process involving algebraic manipulation and calculus to integrate or differentiate as necessary. By solving these, we understand how changes in control affect the evolution of the entire system over time.

Transversality Condition

The Transversality Condition is a boundary condition often necessary for solving optimal control problems. It helps in defining constraints on the costate variables at the terminal time.In this problem, the transversality condition stipulates:- At the final time \( T \), the costate variable \( \lambda(T) = 0 \).This is crucial because \( y(t) \) is free at \( t = T \). Therefore, this condition defines the relationship between the state and costate at the endpoint, providing pivotal information for integrating or solving the differential equations.Together with the given initial condition \( y(0) = y_0 \), the transversality condition ensures the unique determination of the optimal paths for all variables. It essentially closes the circle between the start and end of our analysis, guiding us to the final solution efficiently.