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# 数学代写|MATH414 Optimization Theory

## MATH414课程简介

Introduction to theory and methods for optimization. Topics may include least square analysis, search methods, conjugate direction methods, linear programming, integer programming, and constrained optimization. Pre: 243 or 253A, and 307 or 311; or consent.

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives.[1] It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering[2] to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries

## Prerequisites

In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding “best available” values of some objective function given a defined domain (or input), including a variety of different types of objective functions and different types of domains.

## MATH414 Optimization Theory HELP（EXAM HELP， ONLINE TUTOR）

The tanks $A$ and $B$ shown in Fig. 1-P1 each have a capacity of 50 gal. Both tanks are filled at $t=0, \operatorname{tank} A$ with $60 \mathrm{lb}$ of salt dissolved in water, and $\operatorname{tank} B$ with water. Fresh water enters $\operatorname{tank} A$ at the rate of $8 \mathrm{gal} / \mathrm{min}$, the mixture of salt and water (assumed uniform) leaves $A$ and enters $B$ at the rate of $8 \mathrm{gal} / \mathrm{min}$, and the flow is incompressible. Let $q(t)$ and $p(t)$ be the number of pounds of salt contained in tanks $A$ and $B$, respectively.
(a) Write a set of state equations for the system.
(b) Draw a block diagram (or signal flow graph) for the system.
(c) Find the state transition matrix $\varphi(t)$.
(d) Determine $q(t)$ and $p(t)$ for $t \geq 0$.

(a) $d q(t) / d t=-.16 q(t) ; d p(t) / d t=.16 q(t)-.16 p(t)$
(c) $\varphi_{11}(t)=\epsilon^{-.16 t} ; \varphi_{12}(t)=0 . ; \varphi_{21}(t)=.16 t \epsilon^{-.16 t} ; \varphi_{22}(t)=\epsilon^{-.16 t}$
(d) $q(t)=60 \epsilon^{-16 t} ; p(t)=9.6 t \epsilon^{-16 t}, t \geq 0$.

(a) Write a set of state equations for the mechanical system shown in. Fig. 1-P3. The applied force is $f(t)$, the block has mass $M$, the spring constant is $K$, and the coefficient of viscous friction is $B$. The displacement of the block, $y(t)$, is measured from the equilibrium position with no force applied.
(b) Draw a block diagram (or signal flow graph) for the system.
(c) Let $M=1 \mathrm{~kg}, K=2 \mathrm{~N} / \mathrm{m}, B=2 \mathrm{~N} / \mathrm{m} / \mathrm{sec}$, and determine the state transition matrix $\varphi(t)$.
(d) If $y(0)=0.2 \mathrm{~m}, \dot{y}(0)=0$, and $f(t)=2 \epsilon^{-2 t} \mathrm{~N}$ for $t \geq 0$, determine $y(t)$ and $\dot{y}(t)$ for $t \geq 0$.

(a) $d y(t) / d t=\dot{y}(t) ; d \dot{y}(t) / d t=-K y(t) / M-B \dot{y}(t) / M+f(t) / M$
(c)
\begin{aligned} & \varphi_{11}(t)=\sqrt{2} \epsilon^{-t} \cos (t-\pi / 4) ; \varphi_{12}(t)=\epsilon^{-t} \sin t ; \varphi_{21}(t)=-2 \epsilon^{-t} \sin t \ & \varphi_{22}(t)=\sqrt{2} \epsilon^{-t} \cos (t+\pi / 4) \end{aligned}
(d)
\begin{aligned} & y(t)=2 \sqrt{2} \epsilon^{-t} \cos (t-\pi / 4)+\epsilon^{-2 t}+\sqrt{2} \epsilon^{-t} \cos (t-3 \pi / 4) \ & \dot{y}(t)=-4 \epsilon^{-t} \sin t-2 \epsilon^{-2 t}+2 \epsilon^{-t} \cos t \end{aligned}

Write state equations for the mechanical system in Fig. 1-P5. $\lambda$ is the applied torque, $I$ is the moment of inertia, $K$ is the spring constant, and $B$ is the coefficient of viscous friction. The angular displacement $\theta(t)$ is measured from the equilibrium position with no torque applied.

$d \theta(t) / d t=\dot{\theta}(t) ; d \dot{\theta}(t) / d t=-K \theta(t) / I-B \dot{\theta}(t) / I+\lambda(t) / I$.

Write a set of state equations for the electromechanical system shown in Fig. 1-P7. The amplifier gain is $K_a$, and the developed torque is $\lambda(t)=K_t i_f(t)$, where $K_a$ and $K_t$ are known constants.

$d i_f(t) / d t=-R_f i_f(t) / L_f+K_a e(t) / L_f ; d \omega(t) / d t=K_t i_f(t) / I-B \omega(t) / I$.

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