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Chapter 9: Problem 3

$$ \min _{u(t) \in(-\infty, \infty)} \int_{0}^{1}\left[x(t)+u(t)^{2}\right] d t, \quad \dot{x}(t)=-u(t), \quad x(0)=0, \quad x(1) \text { free } $$

Short Answer

Expert verified
Optimal \( u(t) = \frac{C - t}{2} \). Integrate state equation to find \( x(t) \). Constants \( C \) and \( D \) remain arbitrary.

Step by step solution

01

Set Up the Hamiltonian

The Hamiltonian for the optimal control problem is given by \[ H = x(t) + u(t)^2 + \theta(t)(-u(t)) \], where \( \theta(t) \) is the costate.
02

Determine the Costate Equation

The costate equation is obtained from \[ \frac{d\theta}{dt} = -\frac{\text{d}H}{\text{d}x} = -1. \]So, \( \theta(t) = -t + C \), where \( C \) is a constant.
03

Determine the Optimal Control

To find the optimal control \( u(t) \), solve \[ \frac{\text{d}H}{\text{d}u} = 0. \]This gives \[-2u + \theta = 0 \], hence \( u(t) = \frac{\theta}{2} = \frac{-t + C}{2} \).
04

Integrate the State Equation

From the state equation \( \frac{dx(t)}{dt} = -u(t) \), we get \[ \frac{dx(t)}{dt} = -\frac{-t + C}{2} = \frac{t - C}{2}. \]Integrating, we find \[ x(t) = \frac{t^2}{4} - \frac{Ct}{2} + D, \]where \( D \) is another constant. Apply \( x(0)=0 \) to find \( D=0 \), so \[ x(t) = \frac{t^2}{4} - \frac{Ct}{2} \].
05

Determine the Constants

Since \( x(1) \) is free, we do not have a condition to further determine \( C \) and \( D \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hamiltonian
In optimal control theory, the Hamiltonian plays a critical role. It combines the state, control, and costate variables into one function. This lets us analyze the behavior of the system.

The Hamiltonian for our given problem is: \[ H = x(t) + u(t)^2 + \theta(t)(-u(t)) \] Here, \( x(t) \) represents the state variable, \( u(t) \) is the control variable, and \( \theta(t) \) is the costate variable. The Hamiltonian helps derive the necessary conditions for optimality.

In simple terms, the Hamiltonian is like a dynamic 'score' that captures the balance between minimizing costs and satisfying system dynamics.
Costate Equation
The costate equation is crucial in optimal control because it provides a relationship between the state and the costate variables. It is derived by taking the derivative of the Hamiltonian with respect to the state variable.

In this case: \[ \frac{d\theta(t)}{dt} = -\frac{\text{d}H}{\text{d}x} = -1 \] The solution for this differential equation is: \[ \theta(t) = -t + C \] where \( C \) is a constant of integration.

Having the costate equation helps us understand how the costate variable evolves over time, which supports finding the optimal control strategy.
State Equation
The state equation describes how the state variable evolves over time due to the influence of the control variable. For our problem, the state equation is given by: \[ \frac{dx(t)}{dt} = -u(t) \]

The state equation helps show how changes in the control variable \( u(t) \) affect the state variable \( x(t) \). Integrating the state equation using our optimal control from previous steps, we get: \[ x(t) = \frac{t^2}{4} - \frac{Ct}{2} \] We determine the constant of integration, \( D \), from the given initial condition \( x(0) = 0 \), which simplifies our expression.

This equation shows the trajectory of the state variable over time depending on the applied control.
Control Variable
The control variable \( u(t) \) is the key to optimizing the system. It represents the decision we can adjust to minimize the cost function. To find the optimal control, we differentiate the Hamiltonian with respect to \( u(t) \): \[ \frac{\text{d}H}{\text{d}u} = 0 \ \rightarrow -2u + \theta = 0 \ \rightarrow u(t) = \frac{\theta(t)}{2} \ \rightarrow u(t) = \frac{-t + C}{2} \] This shows that the optimal control depends directly on the costate variable.

Optimal control is crucial for guiding the system towards minimizing the given cost function while obeying the system's dynamics. Understanding how to determine and apply \( u(t) \) is essential for solving optimal control problems.

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Most popular questions from this chapter

Consider the problem \(\max \int_{0}^{1}-u d t, \dot{x}=u^{2}, x(0)=x(1)=0, u \in \mathbb{R}\). (a) Explain why \(u^{*}(t)=x^{*}(t)=0\) solves the problem. (b) Show that the conditions in the maximum principle are satisfied only for \(p_{0}=0\).

(a) Consider the problem $$ \max \int_{0}^{\infty}\left(a x-\frac{1}{2} u^{2}\right) e^{-r t} d t, \quad \dot{x}=-b x+u, x(0)=x_{0}, x(\infty) \text { free, } u \in \mathbb{R} $$ where \(a, r\), and \(b\) are all positive. Write down the current value Hamiltonian \(H^{c}\) for this problem, and determine the system (2). What is the equilibrium point? (b) Draw a phase diagram for \((x(t), \lambda(t))\) and show that for the two solutions which converge to the equilibrium point, \(\lambda(t)\) is a constant. (c) Use sufficient conditions to solve the problem. (d) Show that \(\partial V / \partial x_{0}=\lambda(0)\), where \(V\) is the optimal value function.

$$ \max _{u(t) \in(-\infty, \infty)} \int_{0}^{1}\left[1-u(t)^{2}\right] d t, \dot{x}(t)=x(t)+u(t), \quad x(0)=1, \quad x(1) \text { free } $$

$$ \max _{u \in(-\infty, \infty)} \int_{0}^{10}\left[1-4 x(t)-2 u(t)^{2}\right] d t, \quad \dot{x}(t)=u(t), x(0)=0, \quad x(10) \text { free } $$

Solve the problem (where \(T, \alpha\), and \(\beta\) are positive constants, \(\alpha \neq 2 \beta\) ) $$ \max \int_{0}^{T} e^{-\beta t} \sqrt{u} d t \text { when } \dot{x}(t)=\alpha x(t)-u(t), x(0)=1, x(T)=0, u(t) \geq 0 $$ What happens if the terminal condition \(x(T)=0\) is changed to \(x(T) \geq 0\) ?
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