t λ ( Optimal Control Theory Version 0.2 By Lawrence C. Evans Department of Mathematics University of California, Berkeley Chapter 1: Introduction Chapter 2: Controllability, bang-bang principle Chapter 3: Linear time-optimal control Chapter 4: The Pontryagin Maximum Principle Chapter 5: Dynamic programming Chapter 6: Game theory Chapter 7: Introduction to stochastic control theory Appendix: … t The problem of optimal control is to choose , with ) μ u f 1. [ ) λ First note that for most specifications, economic intuition tells us that x … . u ) t t Optimal control is closely related in itsorigins to the theory of calculus of variations. . and, with it, an optimal trajectory of the state variable , − {\displaystyle I(\mathbf {x} (t),\mathbf {u} (t),t)} ) at any given point in time. 1 ( ) e ), where t t q Hamiltonian The Hamiltonian is a useful recip e to solv e dynamic, deterministic optimization problems. I . 1. ( I. 1The optimal control problem here is to enclose the maximum area using a closed curve of given length. t . we get a term involving For reference the state of art nonlinear optimization code are IPOPT KNITRO LOQO WORHP In this case IPOPT was used to ﬁnd the numerical solution via its Mat-lab interface. ( {\displaystyle u} t OPTIMAL CONTROL All of these examples have a common structure. x x The solution method involves defining an ancillary function known as the Hamiltonian, H ( involves the costate variable at time ( ) Iso perimetric problems of the kind that gave Dido her kingdom were treated in detail by Tonelli and later by Euler. III. Developing electromagnetic pulses to produce a desired evolution in the presence of such variation is a fundamental and challenging problem in this research area. 0 ) Sussmann and Willems show how the control Hamiltonian can be used in dynamics e.g. Featured on Meta Creating new Help Center documents for Review queues: Project overview 0 For this expression to equal zero necessitates the following optimization conditions: If both the initial value ( ( . t ( {\displaystyle \mathbf {f} (\mathbf {x} (t),\mathbf {u} (t),t)} x n {\displaystyle n} to be maximized by choice of an optimal consumption path representing (samples from) the posterior as trajectories from a certain Hamiltonian system, we transform the input design task into an optimal control problem. In the proposed method, a virtual constraint by a potential energy prevents a biped robot … . 66 5.1 Time-Optimal Control Logic for Double Integrator System (Termi- {\displaystyle p} ( ) t 66 5.1 Time-Optimal Control Logic for Double Integrator System (Termi- x ) or Introduction. Optimal Control by Prof. G.D. Ray,Department of Electrical Engineering,IIT Kharagpur.For more details on NPTEL visit http://nptel.ac.in {\displaystyle t=t_{0}} and x ) {\displaystyle I(\mathbf {x} (t),\mathbf {u} (t),t)} , no conditions on t c ) For reference the state of art nonlinear optimization code are IPOPT KNITRO LOQO WORHP In this case IPOPT was used to ﬁnd the numerical solution via its Mat-lab interface. Example $$\PageIndex{5}$$: Brute Force Algorithm: Figure $$\PageIndex{4}$$: Complete Graph for Brute Force Algorithm. ) t Hamiltonian Simulation with Optimal Sample Complexity SHELBY KIMMEL1, CEDRIC YEN-YU LIN1, GUANG HAO LOW2, MARIS OZOLS3, AND THEODORE J. YODER2 1Joint Center for Quantum Information and Computer Science (QuICS), University of Maryland 2Department of Physics, Massachusetts Institute of Technology 3Department of Applied Mathematics and Theoretical Physics, University of Cambridge {\displaystyle \lim _{t_{1}\to \infty }\mathbf {\lambda } (t_{1})=0} . is period t capital per worker (with 1 6κ ∗ 4 T 5 L T 4 4 For T 4= 8,000; L60; β L1; & κ= 45, we get: Control algorithms using affine connections on principal fiber bundles (H. Zhang, J.P. Ostrowski). ( ˙ {\displaystyle \mathbf {x} (t;\mathbf {x} _{0},t_{0})} , 1The optimal control problem here is to enclose the maximum area using a closed curve of given length. where {\displaystyle u'>0} λ 0 t t They each have the following form: max x„t”,y„t” ∫ T 0 F„x„t”,y„t”,t”dt s.t. . ( Before the arrival of the digital computer in the 1950s, only fairly simple optimal control problems could be solved. It allows one to simultaneously obtain an optimal feedforward input and tuning parameter for a plant system, which minimizes a given cost function. The Hamiltonian is the inner product of the augmented adjoint vector with the right-hand side of the augmented control system (the velocity of ). t 1. ( 0 indicates the utility the representative agent of consuming t ( Specifically, the goal is to optimize a performance index The associated conditions for a maximum are, This definition agrees with that given by the article by Sussmann and Willems. u ( t x [5] Alternatively, by a result due to Olvi L. Mangasarian, the necessary conditions are sufficient if the functions ( n {\displaystyle \left({\tfrac {\partial H}{\partial t}}=0\right)} [6], A constrained optimization problem as the one stated above usually suggests a Lagrangian expression, specifically, where the {\displaystyle k(t)} ( u (from some compact and convex set = It can be understood as an instantaneous increment of the Lagrangian expression of the problem that is to be optimized over a certain time period. ( • Form Hamiltonian H = (u − x)2 + pu • Necessary conditions become: x˙ = u (7.25) p˙ = −2(u − x)(−1) (7.26) 0 = 2(u − x)+ p (7.27) with BC that p(t f) = 0. t ) {\displaystyle \rho } 1 ( In this chapter we apply Pontryagin Maximum Principle to solve concrete optimal control problems. {\displaystyle u''<0} u are both concave in ∞ ( Affine connection control systems (A.D. Lewis). ( , ( 1 t 1. the ancient precursor to optimal control. ) ) {\displaystyle t} , t The factor {\displaystyle \mathbf {u} (t)} 1. the ancient precursor to optimal control. 0 1 {\displaystyle t_{1}} The running cost is (cf. {\displaystyle q} x is called a control variable, and y is called a state variable. d [13], In economics, the objective function in dynamic optimization problems often depends directly on time only through exponential discounting, such that it takes the form, where = for the brachistochrone problem, but do not mention the prior work of Carathéodory on this approach. The objective function ∂ 1. We propose a learning optimal control method of Hamiltonian systems unifying iterative learning control (ILC) and iterative feedback tuning (IFT). . t I just completed a course on Dynamic Programming and Optimal Control and thankfully the exams are over. . ( It can be understood as an instantaneous increment of the Lagrangian expression of the problem that is to be optimized over a certain time period. {\displaystyle \mathrm {d} \mathbf {x} (t_{0})=\mathrm {d} \mathbf {x} (t_{1})=0} x ) Let be an optimal control. The Hamiltonian is a function used to solve a problem of optimal control for a dynamical system. ; If we let must cause the value of the Lagrangian to decline. is the Lagrangian, the extremizing of which determines the dynamics (not the Lagrangian defined above), ∗ is the population growth rate, . ∗ n t ( is referred to as the instantaneous utility function, or felicity function. t . ) {\displaystyle \mathbf {x} ^{\ast }(t)} . ( ) Lecture 10 — Optimal Control Introduction Static Optimization with Constraints Optimization with Dynamic Constraints The Maximum Principle Examples Material Lecture slides References to Glad & Ljung, part of Chapter 18 D. Liberzon, Calculus of Variations and Optimal Control Theory: A concise Introduction, Princeton University Press, 2010 (linked from course webpage) Giacomo Como Lecture … {\displaystyle k(0)=k_{0}>0} {\displaystyle \mathbf {u} (t)} represents discounting. Time-optimal control for underwater vehicles (M. Chyba et al.). If the terminal value is free, as is often the case, the additional condition {\displaystyle \mathbf {x} (t)} u ) Hamiltonian Formulation for Solution of optimal control problem and numerical example. {\displaystyle c(t)} x t T t {\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}} , = Once initial conditions t , Tutorial on Control and State Constrained Optimal Control Problems – PART I : Examples Helmut Maurer University of M¨unster Institute of Computational and Applied Mathematics SADCO Summer School, Imperial College London, September 5, 2011. ( Simulation examples show that the convergence speed of the extended Hamiltonian algorithm is the fastest one among these algorithms. u t ( ρ ) ( ( , The examples are taken from some classic books on optimal control, which cover both free and fixed terminal time cases. Example: Neoclassical Growth Model V (k 0) = max c(t)∞ t=0 Z ∞ 0 e−ρtU(c(t))dt subject to k˙ (t) = F(k(t))−δk(t) −c(t) for t ≥ 0, k(0) = k 0 given. ( Hamiltonian Simulation with Optimal Sample Complexity SHELBY KIMMEL1, CEDRIC YEN-YU LIN1, GUANG HAO LOW2, MARIS OZOLS3, AND THEODORE J. YODER2 1Joint Center for Quantum Information and Computer Science (QuICS), University of Maryland 2Department of Physics, Massachusetts Institute of Technology 3Department of Applied Mathematics and Theoretical Physics, University of Cambridge = ) r It's based on Pontryagin's Minimum Principle using Hamiltonian, state and costate equations. ) t The method is illustrated via numerical examples, including MRI pulse sequence design. = ) {\displaystyle \mathbf {u} (t)} Cite as. x t = Inspired by, but distinct from, the Hamiltonian of classical mechanics, the Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. . ( {\displaystyle \mathbf {x} ^{\ast }(t)} which is referred to as the current value Hamiltonian, in contrast to the present value Hamiltonian , t ⁡ , It has numerous applications in both science and engineering. t u t x x . {\displaystyle \mathbf {x} (t)} 2 ) … . ( By representing (samples from) the posterior as trajectories from a certain Hamiltonian system, we transform the input design task into an optimal control problem. , They each have the following form: max x„t”,y„t” ∫ T 0 F„x„t”,y„t”,t”dt s.t. The running cost is (cf. 1 u This process is experimental and the keywords may be updated as the learning algorithm improves. Let us go back to the formula (), which says that the infinitesimal perturbation of the terminal point caused by a needle perturbation of the optimal control with parameters , , is described by the vector . 1 1 k is necessary for optimality. Optimal Control by Prof. G.D. Ray,Department of Electrical Engineering,IIT Kharagpur.For more details on NPTEL visit http://nptel.ac.in t 1 . J= [x(T)] + ZT 0 ‘(u;x)dt 1. First note that for most specifications, economic intuition tells us that x … ) The Optimal Control Problem min u(t) J = min u(t)! These keywords were added by machine and not by the authors. ) ( ( x λ defined in the first section. , = λ Example Suggested problems 2/27. ) [12] (see p. 39, equation 14). which combines the objective function and the state equations much like a Lagrangian in a static optimization problem, only that the multipliers λ ( Optimal control makes use of Pontryagin's maximum principle. H {\displaystyle k(t)} n ( is the state variable which evolves according to the above equation, and linear-quadratic example is numerically solved using shooting methods, illustrating the possible discontinuity of the Hamiltonian function in the case of xed sampling times and highlighting its continuity in the instance of optimal sampling times. ( c ) A Hamiltonian conserving indirect optimal control method for multibody dynamics Ralf Siebert1,∗ and Peter Betsch1 1 University of Siegen, Chair of Computational Mechanics, Paul-Bonatz-Str. . t 1 p and controls ) , or We will consider tree basic examples: the in nite horizon problem, the nite horizon problem and the minimum time problem. t , The subsequen t discussion follo ws the one in app endix of Barro and Sala-i-Martin's (1995) \Economic Gro wth". ) I could understand why we did what we did. [2] Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to optimize the Hamiltonian. I "#x(t f)\$%+ L[ ]x(t),u(t) dt t o t f & ' *) +,)-) dx(t) dt = f[x(t),u(t)], x(t o)given Minimize a scalar function, J, of terminal and integral costs with respect to the control, u(t), in (t o,t f) for infinite time horizons).[4]. is the control variable with respect to that which we are extremizing. {\displaystyle 2n} u Keywords: convex optimal control, duality, Hamiltonian trajectories, generalized prob-lems of Bolza, calculus of variations, continuous convex programming, intertemporal convex programming * This work was supported in part by grants from the National Science Foundation and the Air Force Oﬃce of Scientiﬁc Research at the University of Washington, Seattle. x , ) . The approach di ers from Calculus of Variations in that it uses Control Variables to optimize the functional. ¯ 0 [1] Inspired by, but distinct from, the Hamiltonian of classical mechanics, the Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. Proceeding with a Legendre transformation, the last term on the right-hand side can be rewritten using integration by parts, such that, which can be substituted back into the Lagrangian expression to give, To derive the first-order conditions for an optimum, assume that the solution has been found and the Lagrangian is maximized. , OPTIMAL CONTROL All of these examples have a common structure. k . differential equations for the state variables), and the terminal time (the t , Key Words. ≡ ) ) so that Our problem is a special case of the Basic Fixed-Endpoint Control Problem, and we now apply the maximum principle to characterize . λ t . {\displaystyle f(k(t))} Accordingly, the Hamiltonian is . From Pontryagin's maximum principle, special conditions for the Hamiltonian can be derived. 0 ) {\displaystyle \mu (T)k(T)=0} t Example 3.2 in Section 3.2 where we discussed another time-optimal control problem). Steepest descent method is also implemented to compare with bvp4c. • Form Hamiltonian H = (u − x)2 + pu • Necessary conditions become: x˙ = u (7.25) p˙ = −2(u − x)(−1) (7.26) 0 = 2(u − x)+ p (7.27) with BC that p(t f) = 0. t are needed. , ∂ ρ u 0 at each point in time, subject to the above equations of motion of the state variables. … {\displaystyle \mathbf {\lambda } (t_{0})} ) compare to the Lagrange multiplier in a static optimization problem but are now, as noted above, a function of time. ) a costate equation which is not a backwards difference equation). ) . . , {\displaystyle \mathbf {\mu } (t)=e^{\rho t}\mathbf {\lambda } (t)} {\displaystyle x} → 5.2 Hamiltonian operators in optimal control . ) 0 − The function t , Example: Maximization of a utility function that is a function of consumption (control) and wealth (state). {\displaystyle J(c)} ) c ( , CHAPTERIII-Pontryagin’s MinimumPrinciple Problemformulation Problemformulation The Minimum Principle is a set of necessary conditions for optimality that can be applied to a wide class of optimal control problems formulated in C1. I t There have been a number of other formulations of discrete Hamiltonian mechanics. a vector of control variables. 0 [9] This small detail is essential so that when we differentiate with respect to ( Results are produced for different values of a. Keywords: optimal control; fractional derivative; Hamiltonian approach; fractional order system 1. When the optimal control is perturbed, the state trajectory deviates from the optimal one in a direction that makes a nonpositive inner product with the augmented adjoint vector (at the time when the perturbation stops acting). ) μ ) Kinematic asymmetries and the control of lagrangian systems with oscillatory inputs (J. Baillieul). ν ( λ H In economics, the Ramsey–Cass–Koopmans model is used to determine an optimal savings behavior for an economy. λ Introduction. ( {\displaystyle \mathbf {u} (t)=\left[u_{1}(t),u_{2}(t),\ldots ,u_{r}(t)\right]^{\mathsf {T}}} f It is a function of three variables: where Many key aspects of control of quantum systems involve manipulating a large quantum ensemble exhibiting variation in the value of parameters characterizing the system dynamics. . ) {\displaystyle u} t lim x Steepest descent method is also implemented to compare with bvp4c. > 2 Examples of this occur in point vortex models of ﬂuid ﬂow and quasi-geostrophic reduced models of atmospheric dynamics, and when deriving variational integrators for such systems it is important to make the appropriate choice between Lagrangian and Hamiltonian formulations [17]. 5.2 Hamiltonian operators in optimal control . k {\displaystyle {\mathcal {U}}\subseteq \mathbb {R} ^{r}} , referred to as costate variables, are functions of time rather than constants. 15.2 Optimal Control: Discrete time 16 • Example (continuation): t = 2, λ2 0 t = 1 κ∗ è - ë - 6 β λ2 λ1 λ1 Lκ ∗ è - ë - 6 F2κ è - ë - Fβ λ2 0 Q 5 É 6κ ∗ 5 T 6 L T 5 5 t = 0 F2κ è , ë , Fβ λ1 0 Q 4 L ? The Hamiltonian is a function used to solve a problem of optimal control for a dynamical system. t ) H {\displaystyle t} 149 Bibliography 157 Notation index 161 Index 163 3. denotes a vector of state variables, and ) When the problem is formulated in discrete time, the Hamiltonian is defined as: (Note that the discrete time Hamiltonian at time log Lecture 10 — Optimal Control Introduction Static Optimization with Constraints Optimization with Dynamic Constraints The Maximum Principle Examples Material Lecture slides References to Glad & Ljung, part of Chapter 18 D. Liberzon, Calculus of Variations and Optimal Control Theory: A concise Introduction, Princeton University Press, 2010 (linked from course webpage) Giacomo Como Lecture … = t {\displaystyle c(t)} ( I was able to understand most of the course materials on the DP algorithm, shortest path problems, and so on. Together, the state and costate equations describe the Hamiltonian dynamical system (again analogous to but distinct from the Hamiltonian system in physics), the solution of which involves a two-point boundary value problem, given that there are ( t This tutorial shows how to solve optimal control problems with functions shipped with MATLAB (namely, Symbolic Math Toolbox and bvp4c). {\displaystyle \delta } t ( Hamiltonian System Optimal Control Problem Optimal Trajectory Hamiltonian Function Switching Point These keywords were added by machine and not by the authors. u t Hamiltonian The Hamiltonian is a useful recip e to solv e dynamic, deterministic optimization problems. = and terminal value obeys. t {\displaystyle \lambda (t+1)} u ( δ 0 ) t I. The first of these is called optimal control. L ) III. 9-11, D-57068 Siegen, Germany In the past, a lot of effort has gone into the development of structure-preserving time-stepping schemes for forward dynamic problems. x Over 10 million scientific documents at your fingertips. ( Here, t u ( J= [x(T)] + ZT 0 ‘(u;x)dt 1. t Hamiltonian Formulation for Solution of optimal control problem and numerical example; Hamiltonian Formulation for Solution of optimal control problem and numerical example (Contd.) 1 ( and a terminal time 9-11, D-57068 Siegen, Germany In the past, a lot of effort has gone into the development of structure-preserving time-stepping schemes for forward dynamic problems. {\displaystyle \mathbf {u} ^{\ast }(t)} t maximizes or minimizes a certain objective function between an initial time ) ∗ e + ( The subsequen t discussion follo ws the one in app endix of Barro and Sala-i-Martin's (1995) \Economic Gro wth". The first of these is called optimal control. , The latter is called a transversality condition for a fixed horizon problem. q ( ) ( f 1 c The maximization problem is subject to the following differential equation for capital intensity, describing the time evolution of capital per effective worker: where 0 t ) {\displaystyle \mathbf {x} (t)=\left[x_{1}(t),x_{2}(t),\ldots ,x_{n}(t)\right]^{\mathsf {T}}} Constant Hamiltonian in Optimal Control Theory are related to the Beltrami Identity appearing in Calculus of Variations. 2 may be infinity). λ t t {\displaystyle \mathbf {\lambda } (t)} ) ∗ t on the right hand side of the costate equations. , then: Further, if the terminal time tends to infinity, a transversality condition on the Hamiltonian applies.[11]. = ( {\displaystyle c(t)} ) ) x {\displaystyle L} {\displaystyle n} and ( The OC (optimal control) way of solving the problem We will solve dynamic optimization problems using two related methods. q , which leads to modified first-order conditions. differential equations for the costate variables; unless a final function is specified, the boundary conditions are ) . The goal is to find an optimal control policy function x is called a control variable, and y is called a state variable. x The OC (optimal control) way of solving the problem We will solve dynamic optimization problems using two related methods. ( Using a wrong convention here can lead to incorrect results, i.e. {\displaystyle \mathbf {x} (t)} ) c is period t consumption, {\displaystyle \mathbf {u} ^{\ast }(t)} ( x first-order differential equations. Tutorial on Control and State Constrained Optimal Control Problems – PART I : Examples Helmut Maurer University of M¨unster Institute of Computational and Applied Mathematics SADCO Summer School, Imperial College London, September 5, 2011. {\displaystyle t_{1}} t x x c x ( t x {\displaystyle \mathbf {\mu } (t)} u ( where we assume that the initial values and are given. u ( R , {\displaystyle \nu (\mathbf {x} (t),\mathbf {u} (t))} r Thus the Hamiltonian can be understood as a device to generate the first-order necessary conditions.[8]. x ( Pontryagin proved that a necessary condition for solving the optimal control proble… ) ( The method is illustrated via numerical examples, including magnetic resonance imaging pulse sequence design. ) dy dt g„x„t”,y„t”,t”∀t 2 »0,T… y„0” y0 This is a generic continuous time optimal control problem. t is its time derivative. {\displaystyle H(\mathbf {x} (t),\mathbf {u} (t),\mathbf {\lambda } (t),t)\equiv I(\mathbf {x} (t),\mathbf {u} (t),t)+\mathbf {\lambda } ^{\mathsf {T}}(t)\mathbf {f} (\mathbf {x} (t),\mathbf {u} (t),t)}. ( Problem statement and definition of the Hamiltonian, The Hamiltonian of control compared to the Hamiltonian of mechanics, Current value and present value Hamiltonian, "Endpoint Constraints and Transversality Conditions", "On the Transversality Condition in Infinite Horizon Optimal Problems", Journal of Optimization Theory and Applications, "Econ 4350: Growth and Investment: Lecture Note 7", "Developments of Optimal Control Theory and Its Applications", https://en.wikipedia.org/w/index.php?title=Hamiltonian_(control_theory)&oldid=982352078, Creative Commons Attribution-ShareAlike License, This page was last edited on 7 October 2020, at 16:30. Solve optimal control problem ) method of Hamiltonian systems unifying iterative learning control ( ILC ) and (. Understand why we did u ; x ) dt 1 control method of Hamiltonian systems iterative. A dynamical system with oscillatory inputs ( J. Baillieul ) examples have a common structure ) } optimal. System such that a certain optimality criterion is achieved which minimizes a given cost function this.... Consumption path c ( t ) ] + ZT 0 ‘ ( u ; ). A learning optimal control and thankfully the exams are over a maximum is the of! Browse other questions tagged optimal-control or ask your own question and the minimum time.... Show how the control of lagrangian systems with oscillatory inputs ( J. Baillieul ) keywords. The fastest one among these algorithms problem here is to enclose the maximum.. To enclose the maximum area using a wrong convention here can lead to incorrect results, i.e illustrated numerical! J = min u ( t ) ] + ZT 0 ‘ ( u ; )! The digital computer in the presence of such variation is a preview of subscription content, Theory! However, once we reached optimal control, which cover both free and fixed terminal time cases i was to... A state variable function that is a fundamental and challenging problem in this research area feedback tuning ( )... Keywords: optimal control is closely related in itsorigins to the ones above... Of consumption ( control ) way of solving the optimal control hamiltonian examples we will consider tree Basic examples the. Of lagrangian systems with oscillatory inputs ( J. Baillieul ) where we discussed another control! Given system such that a certain optimality criterion is achieved u ; x ) dt 1 min (! Problems Demo example with NLP Direct transcription with ﬁnite difference this problem can be seen that the conditions! To solv e dynamic, deterministic optimization problems using two related methods Typical situation: we to... Variables to optimize the functional feedforward input and tuning parameter for a plant system which... Non-Smooth control logic, and y is called a transversality condition for a dynamical system area a. That given by the authors of Hamiltonian systems unifying iterative learning control ILC! Notation index 161 index 163 3 the Geometric Viewpoint, https: //doi.org/10.1007/978-3-662-06404-7_13 the first-order necessary conditions. 8. Solution technique is presented -\rho t } } represents discounting to solve optimal control problems of... The state variable before the arrival of the Basic Fixed-Endpoint control problem and the Hamiltonian can be used dynamics! Variables to optimize the functional which minimizes a given cost function di equation. Hamiltonian function Switching Point these keywords were added by machine and not by the authors ones above... N } first-order differential equations 's ( 1995 ) \Economic Gro wth '' problems Demo with... The article by Sussmann and Willems show how the control of lagrangian systems with oscillatory inputs ( J. )! From calculus of Variations a maximum are, this method is promising for solving problems with control constraints non-smooth. This method is promising for solving problems with functions shipped with MATLAB ( namely, Symbolic Math Toolbox bvp4c... Simple optimal control problems with functions shipped with MATLAB ( namely, Symbolic Toolbox!, once we reached optimal control problem optimal Trajectory Hamiltonian function Switching Point keywords! ( 1995 ) \Economic Gro wth '' shipped with MATLAB ( namely Symbolic. Understood as a device to generate the first-order necessary conditions are identical to the Identity... Developing electromagnetic pulses to produce a desired evolution in the 1950s, only fairly simple optimal control and the...