Exercises - Optimal Control

Coupled vector Euler–Lagrange equations
Consider
$J[x,y] = \int_0^T \left[ \frac{1}{2} \left( \dot{x}^2+\dot{y}^2 \right) + \alpha x\dot{y} - \frac{\omega^2}{2} \left( x^2+y^2 \right) \right] \,\mathrm{d}t,$
(1)
subject to
$x(0)=x_0, \qquad y(0)=y_0, \qquad x(T)=x_f, \qquad y(T)=y_f.$
(2)
(a) Derive the Euler–Lagrange equation for $x(t)$ .
(b) Derive the Euler–Lagrange equation for $y(t)$ .
(c) Write the resulting equations in the matrix form
$\ddot{\boldsymbol{q}} + A\dot{\boldsymbol{q}} + B\boldsymbol{q} = \boldsymbol{0}, \qquad \boldsymbol{q} = \begin{bmatrix} x\\y \end{bmatrix}.$
(3)
(d) For $\alpha=0$ , solve the complete boundary-value problem.
Weighted minimum-slope trajectory
Determine the extremal of
$J[x] = \frac{1}{2} \int_0^T e^{\beta t}\dot{x}^2(t)\,\mathrm{d}t$
(4)
subject to
$x(0)=x_0, \qquad x(T)=x_f.$
(5)
(a) Derive the Euler–Lagrange equation.
(b) Solve explicitly for $x^*(t)$ .
(c) Evaluate $J[x^*]$ .
(d) Show that the solution approaches the straight-line path as
$\beta\rightarrow 0.$
(6)
Free terminal state
Consider
$J[x] = \int_0^T \left( \dot{x}^2+\omega^2x^2 \right) \,\mathrm{d}t,$
(7)
subject to
$x(0)=x_0,$
(8)
where $T$ is fixed and $x(T)$ is free.
(a) Derive the Euler–Lagrange equation.
(b) Derive the natural boundary condition at $t=T$ .
(c) Solve the resulting boundary-value problem.
(d) Determine $x(T)$ explicitly in terms of $x_0$ , $T$ , and $\omega$ .
Free final time
Consider
$J[x,t_f] = \int_0^{t_f} \left[ 1+ \frac{1}{2} \left( \dot{x}-at \right)^2 \right] \,\mathrm{d}t,$
(9)
subject to
$x(0)=0, \qquad x(t_f)=x_f,$
(10)
where $t_f$ is free.
(a) Derive the Euler–Lagrange equation.
(b) Determine the general extremal $x(t)$ .
(c) Derive the free-final-time condition.
(d) Reduce the unknown final time to a scalar algebraic equation.
(e) State how all positive admissible roots should be checked.
Mass–spring boundary-value problem and singular terminal times
For
$L = \frac{1}{2}m\dot{x}^2 - \frac{1}{2}kx^2,$
(11)
consider the endpoint conditions
$x(0)=x_0, \qquad x(T)=x_f.$
(12)
(a) Derive the Euler–Lagrange equation.
(b) Solve the differential equation for $m>0$ and $k>0$ .
(c) Determine the constants in the general solution.
(d) Show that a unique solution exists when
$\sin\left(\sqrt{\frac{k}{m}}\,T\right)\neq 0.$
(13)
(e) Analyze separately the case
$\sin\left(\sqrt{\frac{k}{m}}\,T\right)=0$
(14)
and determine when the boundary-value problem has no solution or infinitely many solutions.
First variation and the natural boundary condition
Let
$J[x] = \int_{t_0}^{t_f} L(x,\dot{x},t)\,\mathrm{d}t,$
(15)
where $t_0$ , $t_f$ , and $x(t_0)$ are fixed, but $x(t_f)$ is free.
Starting from
$x_\epsilon(t) = x(t)+\epsilon\eta(t),$
(16)
derive the complete first variation. Use integration by parts to show that stationarity requires
$\frac{\partial L}{\partial x} - \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial L}{\partial\dot{x}} \right) = 0$
(17)
and
$\left. \frac{\partial L}{\partial\dot{x}} \right|_{t_f} = 0.$
(18)
Endpoint constrained to a curve
Let the initial endpoint be fixed and let the terminal endpoint satisfy
$\psi(x_f,t_f) = x_f^2+c^2t_f^2-R^2 = 0.$
(19)
(a) Starting from
$\delta x_f = \delta x(t_f) + \dot{x}(t_f)\delta t_f,$
(20)
derive the linearized endpoint constraint
$\psi_x\,\delta x_f + \psi_t\,\delta t_f = 0.$
(21)
(b) Compute $\psi_x$ and $\psi_t$ .
(c) Express $\delta x(t_f)$ in terms of $\delta t_f$ , assuming $\psi_x\neq 0$ .
(d) Substitute this relation into the terminal contribution
$\left. \frac{\partial L}{\partial\dot{x}} \right|_{t_f} \delta x(t_f) + L(t_f)\delta t_f$
(22)
and derive the corresponding scalar transversality relation.
Planar locus-to-locus formulation
Let
$\boldsymbol{x}(t) = \begin{bmatrix} x(t)\\y(t) \end{bmatrix}.$
(23)
The initial state must lie on a circle of radius $r_0$ centered at $(a_0,b_0)$ , and the final state must lie on a circle of radius $r_f$ centered at $(a_f,b_f)$ .
(a) Write the two scalar endpoint constraints.
(b) Collect them in the vector form
$\boldsymbol{\phi} \bigl( \boldsymbol{x}(t_0), t_0, \boldsymbol{x}(t_f), t_f \bigr) = \boldsymbol{0}.$
(24)
(c) Derive the first-order variations of both endpoint constraints.
(d) Identify the normal vector to each circle at the corresponding endpoint.
General fixed point-to-point boundary mapping
Express
$\boldsymbol{x}(t_0)=\boldsymbol{x}_0, \qquad \boldsymbol{x}(t_f)=\boldsymbol{x}_f$
(25)
in the form
$\boldsymbol{\phi} \bigl( \boldsymbol{x}(t_0), t_0, \boldsymbol{x}(t_f), t_f \bigr) = \boldsymbol{0}.$
(26)
Then:
(a) determine the dimension of $\boldsymbol{\phi}$ when $\boldsymbol{x}\in\mathbb{R}^n$ ;
(b) compute the Jacobians of $\boldsymbol{\phi}$ with respect to $\boldsymbol{x}(t_0)$ and $\boldsymbol{x}(t_f)$ ;
(c) write the first-order variation $\delta\boldsymbol{\phi}$ ;
(d) show that fixed endpoint states imply $\delta\boldsymbol{x}_0=\boldsymbol{0}$ and $\delta\boldsymbol{x}_f=\boldsymbol{0}$ .
Moving upper limit and free-final-time condition
Consider
$J[x,t_f] = \int_{t_0}^{t_f} L(x,\dot{x},t)\,\mathrm{d}t,$
(27)
with fixed $t_0$ , fixed $x(t_0)$ , fixed terminal value $x(t_f)=x_f$ , and free $t_f$ .
(a) Use the Leibniz rule to show that variation of the upper limit contributes
$L(t_f)\delta t_f.$
(28)
(b) Derive the complete first variation.
(c) Use
$\delta x(t_f) = -\dot{x}(t_f)\delta t_f$
(29)
to eliminate the terminal trajectory variation.
(d) Derive
$\left[ L - \dot{x} \frac{\partial L}{\partial\dot{x}} \right]_{t_f} = 0.$
(30)
(e) Show that this condition is equivalent to
$H(t_f)=0, \qquad H = \dot{x} \frac{\partial L}{\partial\dot{x}} - L.$
(31)