# 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)

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### Find a field of extremals

I have this exercise that we have done in class where I have problem understanding the solution. We consider the following optimization problem $\int^{2}_{1} y'(x) + x^2 y'(x)^2 dx$. Calculate a ...
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Let a discrete time system be \begin{align} x[k+1]=Ax[k] \end{align} If the system in Eq.1 is stable then always it will satisfy the Lyapunov equation as described below. Let the Lyapunov function be $... 1 vote 1 answer 43 views ### Question on nonlinear optimal control problem Problem: Given a discrete equation of state $$x^{k+1} = x^k -u^k,\ u^k \ge 0.$$ Our goal is to drive in$N$steps the system to the origin, from$x^0$and minimize the cost function $$J(x,u) := \sum_{... • 707 0 votes 0 answers 23 views ### epsilon balls and 0- and 1- norms in optimal control Please consider the following excerpt from Calculus of Variations and Optimal Control Theory, A Concise Introduction by Daniel Liberzon Here the space \mathcal{C}^k is the space of k times ... • 974 0 votes 0 answers 26 views ### How to solve the optimal control problem? Optimal control problem: \dot{x}=rx-\alpha u J=\int_{0}^{\infty}e^{-\rho t}((b-\frac{u}{2})u-cx)dt, \mapsto Max Where r,\alpha, b,c, \rho \in R and all parameters are not negative. where as ... • 74 0 votes 0 answers 16 views ### Observability of plates - Proof of the improvement (condition Observability) hey guys I was learning about the Observability of plates and I came across integral equality I didn't get how they got into it from the integral here is the ... 0 votes 0 answers 16 views ### why does maximized utility of consumption(merton problem) exist? Agent controls his proportion of wealth invested in the stock \alpha_t and his consumption rate c_t. Dynamic of wealth: dX_t=X_t[(\alpha_t(u-r)+r)dt+\alpha_t \sigma_t dW_t]-c_t dt value ... 0 votes 0 answers 47 views ### Is L^{\infty}([0, 1]; \mathbb{R}^{n}) densely embedded in some Hilbert space? [duplicate] I'm solving a control problem and I have a question. Is it possible to show that L^{\infty}([0, 1]; \mathbb{R}^{n}) is densely embedded in some Hilbert space? 0 votes 0 answers 35 views ### The conditions of optimization task Could you please help me to solve the task using optimization method? The factory produces three types of glue. And four types of chemicals are used for its production: starch, gelatin, alum and chalk.... • 63 0 votes 0 answers 15 views ### Rank condtion and stationary point for constrained systems I am trying to understand how is Hamiltonian function defined. For the optimization problem, \min L(x,u) s.t. f(x,u)=0, a necessary condition for a minimum is that \left[\begin{array}{c}d L \\ d f\... • 87 -1 votes 2 answers 27 views ### Solving a diffrential equation with deriviative of more than one dependent variable How should I go about solving a differential equation of the form: \frac {d}{dR}(f_1(R)g_1(R)+f_2(R) g_2 (R))=0 where f_1(R) and f_2(R) are known. I am trying to solve for g_1(R) and g_2(R)... 1 vote 1 answer 122 views ### Solution to an algebraic Riccati equation with complex matrices I am trying to find the analytical solution for the following Riccati equation:$$ 0 = F + W^\dagger P(t) + P(t) W + P(t)X P(t). $$In my particular problem I know that it has a solution. In this ... • 125 1 vote 0 answers 56 views ### Solve discrete Algebratic Riccati Equation if S is non-square - How? I have the state-space model$$\dot x = Ax + Bu + Wy = Cx + Du + E$$where$E\in \mathbb{R}^{n \times (N-1)}$is a noise vector and$W\in \mathbb{R}^{p \times (N-1)}$a disturbance vector. What I ... • 2,732 2 votes 0 answers 33 views ### Viscosity supersolution of HJB equation I'm having some troubles solving a question in an exercise. The set-up is the following: Let$\sigma:\mathbb R^d \rightarrow \mathbb R^{d\times d}$be a$\mathcal C^2$map which is bounded and has ... • 21 0 votes 1 answer 42 views ### What is the ratio of a transfer function$G(s)$? If I have a transfer function$G(s) = \frac{Y(s)}{U(s)}$where the$G(s)$is the ratio between the amplitude of$Y(s)$and$U(s)$. In what unit is the ratio? Let's say that we have a input signal$u(t)...
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Static Game Let $i \in \{1, 2\}$ denote a player. Each player can execute an action $a_i \in A_i$, where $A_i \subseteq \mathbb R$ denotes the set of feasible actions. Given a pair of actions \$a = (...