Question

Find the optimal paths of the control, state, and costate variables that will Maximize \(\int_{0}^{2}\left(2 y-3 u-a u^{2}\right) d t\) subject to \(\quad y^{\prime}=u+y\) and \(\quad y(0)=5 \quad y(2)\) free

Short answer

Use Pontryagin's Maximum Principle to find \( u(t) = \frac{\lambda(t) + 3}{2a} \), then solve resulting differential equations.

Step-by-step solution

Understand the Problem

We are given a problem of optimal control where we need to maximize the integral of a function over a given interval. The function includes state variable \( y \), control variable \( u \), and a quadratic term in \( u \). The dynamic constraint is given by the differential equation \( y' = u + y \), with an initial condition \( y(0) = 5 \) and endpoint \( y(2) \) left free. We will need to identify the optimal path for each involved variable.

Set Up the Hamiltonian

The Hamiltonian, \( H \), combines the objective function and the dynamics. For our problem, \( H = 2y - 3u - au^2 + \lambda(t)(u + y) \), where \( \lambda(t) \) is the costate variable that weighs the importance of the state constraint.

Derive the Necessary Conditions

To solve the optimal control problem, we apply the Pontryagin's Maximum Principle which imposes the conditions: 1. \( \frac{dH}{du} = 0 \) gives \( -3 - 2au + \lambda(t) = 0 \). Solve for \( u \). 2. State dynamics: \( y'(t) = u + y \).3. Costate dynamics: \( \lambda'(t) = -\frac{dH}{dy} = -2 - \lambda(t) \).

Solve the First-Order Condition for Control Variable

From \( -3 - 2au + \lambda(t) = 0 \), solve for \( u \): \[ u(t) = \frac{\lambda(t) + 3}{2a} \]. This gives the expression for the control in terms of the costate.

Solve the Differential Equations for \( y(t) \) and \( \\lambda(t) \)

Using \( y'(t) = u + y \), replace \( u \) with the expression from Step 4: \[ y'(t) = \frac{\lambda(t) + 3}{2a} + y \]. Simultaneously solve \( \lambda'(t) = -2 - \lambda(t) \).

Apply Boundary Conditions

The boundary condition at the beginning \( y(0) = 5 \) must be applied to the solution of \( y(t) \). The transversality condition \( \lambda(2) = 0 \) is applied since \( y(2) \) is free. This will provide any remaining constants from the solution.

Find the Optimal Paths

Integrate the solutions from Step 5 using values derived from Step 6. Determine specific paths for \( y(t) \), \( u(t) \), and \( \lambda(t) \), ensuring that initial and final conditions are met in the process.

Key concepts

Hamiltonian

In optimal control theory, the Hamiltonian plays a crucial role as it combines the system's dynamics with the objective function we aim to optimize. In essence, it provides a complete picture of the system's evolution over time. For an optimal control problem, like the one given:

• The Hamiltonian, denoted as \( H \), will often include terms for state variables, control variables, and their derivatives.
• In our example, the Hamiltonian is formed by combining the objective function terms \((2y - 3u - au^2)\) with the dynamic constraint \(\lambda(t)(u + y)\).
• Here, \( \lambda(t) \) is the costate variable, which weighs the constraint's importance.

By understanding and deriving the Hamiltonian, we can set up necessary conditions to find optimal control paths. It is essential because it provides insight into how changes in control or state affect the overall objective.

Pontryagin's Maximum Principle

Pontryagin's Maximum Principle is a cornerstone of optimal control theory. It provides the necessary conditions to identify the best possible control strategy to optimize the desired outcome.The Maximum Principle imposes conditions to help find the optimal control \( u(t) \):

• We derive conditions using the Hamiltonian's partial derivatives. For example, setting \(\frac{dH}{du} = 0\) gives critical insights into the optimal control's behavior.
• These conditions help balance the trade-off between immediate rewards and future benefits, which is often captured by the control and state variables dependency.

By applying this principle, we solve for \( u \) as seen in the solution process: \[ u(t) = \frac{\lambda(t) + 3}{2a} \].This relationship indicates how control strategies should adjust based on other system variables to maintain optimal performance.

Differential Equations

Differential equations are fundamental to describing dynamic systems in optimal control problems. They define how the state variables change over time given certain control inputs and initial conditions.For our specific problem:

• We have a state equation \( y'(t) = u + y \), representing how the state \( y \) evolves over time.
• The costate equation \( \lambda'(t) = -2 - \lambda(t) \) tracks the dynamics of \( \lambda(t) \), providing essential feedback on state changes.

Both equations need to be simultaneously solved to ensure that both the control and the state follow their optimal paths. Differential equations provide a path towards understanding the dynamic interplay of these variables throughout the optimization interval. By incorporating initial and boundary conditions, like \( y(0) = 5 \), these solutions become specific and relevant to practical scenarios.

Costate Variable

The costate variable, often denoted as \( \lambda(t) \), is a part of the optimal control framework critical to linking constraints with the objective function. It functions similar to Lagrange multipliers in static optimization problems.Key aspects of the costate variable include:

• It represents the shadow price of state constraints, offering insight into how much the objective would change with a slight alteration in the state constraints.
• In our problem, the evolution of \( \lambda(t) \) is dictated by the costate differential equation: \( \lambda'(t) = -2 - \lambda(t) \).
• The terminal condition, \( \lambda(2) = 0 \), ensures that if the final state is free, the shadow price is zero at this boundary point, ensuring optimality at the endpoint.

Interpreting \( \lambda(t) \) provides a deeper understanding of the state variable's influence on achieving the optimal objective over time.