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Questions tagged [optimal-control]

Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. (Def: http://en.m.wikipedia.org/wiki/Optimal_control)

Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. Reference: Wikipedia.

The method is largely due to the work of Lev Pontryagin and Richard Bellman.

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Inseparable terminal and running cost in optimal control problems?

I have the following version of time optimal control problem for a two dimensional system with terminal equality and state inequality constraint. \begin{align} \mathbb{J}(u)= (T-\int_{0}^{t_f}\mathrm{d}t)^2,\;\;\;\;\; T =…
storm_chaser
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Optimal Control: What if state depends on control $x(t,u)$?

I'm from econ. That is I'm not conceptually familiar with the underlying math of dynamical systems. When I usually deal with dynamic systems it's of the form $$ \max_u\int{F(t,x(t),u(t))}\\ \text{s.t.}~\dot{x}(t)=f(t,x(t),u(t))\\ x(0)=x_0 $$ where…
Simon
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optimal controller beginner

In my optimal control course, we have done problems such as: Find the optimal trajectory $x^*(t)$ for minimizing: $J = \int_0^{tf} \left[ \frac{1}{2}\dot{x}^2(t) + x(t)\dot{x}(t) + \dot{x} \right] dt$ with different combinations of: fixed…
user2913869
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Optimal control and Value function

Let's consider this optimal control problem: Minimize $-x(1)$, subject to $dx(t)/dt=x(t)u(t)$ for almost every $t \in [0,1]$, $x(0)=0$ among all the admissible controls $u:[0,1] \to [0,1]$ such that $u$ is Lebesgue measurable. How can I compute the…
effezeta
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Understanding one condition in Pontryagin Maximum Principle

This is a public course note about the optimal control problem. https://math.berkeley.edu/~evans/control.course.pdf There is one condition that I do not get. Is there any intuition that we need "the mapping is constant"? And how does that be…
ftor
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Doubt - Optimal Control Theory Problem with transversality condition

I'm trying to solve this optimal control theory problem, but i'm having some problem finding the constants $k_1$ and $k_2$. I will show what I have developed so far. The problem: $$V(y)=\int_{0}^{1}u^2+2yu+2y^2\,dt$$ with $$y'=y+u;\quad y(0)=0;\quad…
Jackaba
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Please help to solve this Riccati equation

I am a newbie in control area. I get stuck at an Riccati equation. I find everywhere and I can't find an answer. A brief description would like, Say I have an equation: $P = Q + \zeta^TPA$ and $\zeta = A^T (I + PBR^{-1} B^T)^{-1}$ all of the…
louis
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Bellman's Principle of Optimality

I'm currently reading Pham's Continuous-time Stochastic Control and Optimization with Financial Applications however I'm slightly confused with the way the Dynamic Programming Principle is presented. In particular, the Theorem is stated in terms of…
mark
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How to solve an optimal control problem where one variable is "reset" at time $T$?

Standard problem. Say we have an optimal control problem with the following state variables $$ \begin {align} &\min_{u_t} \int_0^\infty C(t,x_t)\,dt, \text { subject to } \\ &\dot x_t=f(x_t,y_t,u_t)\\ &\dot y_t = g(u_t)\\ &0\leq u_t\leq…
user56834
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how to find a reachable set?

I have this example: Let $n=2$ and $m=1$, $A=[-1,1]$, and write $x(t)=(x_1,x_2)^T$. Suppose $ \left \{ \begin{matrix} x'_1=0 \\ x'_2=\alpha(t) \end{matrix} \right.$ This is the system of the form $x'=Mx+N\alpha$,…
Alex Pozo
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No. of control and state variables in optimal control theory

I am given a question in which I can identify 2 control variables and 1 state variable. Can I check if I can still apply Hamiltonian here to solve this question? I have only dealt with questions where the number of control variables is the same as…
Sue Qin
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Lyapunov equation for discrete systems - LQR

I have following dicrete-time system $$ x(k+1) = A.x(k) + B.u(k) \\ y(k) = C.x(k) $$ I am supposed to find controller in form of $$ u^* = -K.x $$so that following performance index is minimized $$ \sum\limits_{k=0}^\infty\frac1{2} x^T Q x + u^T R u…
Martin G
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Minimum containers of given volumes for packing whole volume of a liquid

Suppose there are containers with volumes(integers) $V1, V2, \cdots, VN$ where $V1 =1$, and rest of the containers are distinct but not necessarily consecutive. Each container can be available/used more than once for filling liquids. Is there a…
user1191177
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Maximize an integral under some conditions

I should maximize the following integral (by determine the optimal time $t_o$) $C(u)=\int_0^T e^{-rt}(1-u(t))x(t)dt$ with $u(t)=1$ if $t0$. $0
Brian
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linear quadratic regulator proof

Given a basic LQR regulator problem to design an optimal control, $u^*(t)$ for the state-space system: $\dot{x} = Ax + Bu$ which minimizes the following functional: $J = \frac{1}{2}\int_{0}^{t_f} \left[ x^TQx + u^TRu \right]dt$ I am to prove that…
user2913869
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