endobj 0000008684 00000 n
Introduction to Optimal Control 1.1 Some examples Example 1.1.1. 0000003941 00000 n
�`@Ex1��Wsy�̝֦a' ~@�+Iy9�ߚ�A�4�L=Q��Nʠ}�(����&ud7/�
�DqRў��K��q���{|c���. (2004) Examples of Optimal Control Problems. 0000041535 00000 n
the control of the decision-maker (any value ut ∈Ut may be chosen). J. of Economics 91 (1), 1 75- 1 78,1 989 Sierstad, Atle and Sydsaeter, Knut: Optimal Control Theory with EconomicApplications. endobj 0000062702 00000 n
0000007526 00000 n
0000013744 00000 n
In Section 1, we introduce the denition of Optimal Control problem and give a simple example. Econ 431: Bang-Bang Optimal Control Example Example 1 Find the optimal control that will Max V= R2 0 (2y−3u)dt subject to y0= y+u y(0) = 4 y(2) free and u(t) ∈U=[0,2] Since the problem is characterized by linearity in uand a closed control set, we can expect boundary solutions to occur. H��T�n�0��+t��DR%`͡�0��N���k4E�Y�~��m�,�I�-v0 ��{�㣜�o���aZ�͇�G2h�U��7���J���1���@���U�֤P�\�\���#O�I#���t�HqI�\���_m���q�Y�l�9�u��M{�_�� �
6H9Ѿp̕�e�|��$��~YH[�����g�B��#2x�zuP@R�u8R{��{���7� 2�3�7�A�A����Yi�_4 0000028901 00000 n
0000024720 00000 n
9 0 obj << /S /GoTo /D [30 0 R /Fit ] >> Scand. 0000057834 00000 n
It's a way of solving an optimization problem in continuous time. For example, in economic models, negative values may be infeasible. Optimal Control Direct Method Examples version 1.0.0.0 (47.6 KB) by Daniel R. Herber Teaching examples for three direct methods for solving optimal control problems. We will start by looking at the case in which time is discrete (sometimes called 0000051261 00000 n
0000003103 00000 n
The OC (optimal control) way of solving the problem We will solve dynamic optimization problems using two related methods. 0000039981 00000 n
: The report presents an introduction to some of the concepts and results currently popular in optimal control theory. endstream
endobj
37 0 obj
<>
endobj
38 0 obj
<>
endobj
39 0 obj
<>stream
0000011726 00000 n
0
0000009372 00000 n
John Maynard Keynes published a book in 1936 called The General Theory of Employment, Interest, and Money, laying the groundwork for his legacy of the Keynesian Theory of Economics.It was an interesting time for economic speculation considering the dramatic adverse effect of the Great Depression. %PDF-1.4 (The Maximum Principle) 0000013960 00000 n
Optimal control theory will serve as the basis to arrive at economic rules for reaching desired goals, such as giving people the most enjoyment possible. • then there is at least an optimal path for the state variable x ∗≡ {x 0,x1,...} The most common dynamic optimization problems in economics and ﬁnance have the following common assumptions • timing: the state variable xt is usually a stock and is measured at the beginning of period t and the control ut is usually a ﬂow and is measured 0000062258 00000 n
The purpose of the article was to derive the technique for solving optimal control problems by thinking through the economics of … 79 0 obj
<>stream
%%EOF
h�T�Mo�0��� Rational behavior refers to a decision-making process that is based on making choices that result in an optimal level of benefit or utility. AGEC 642 Lectures in Dynamic Optimization Optimal Control and Numerical Dynamic Programming Richard T. Woodward, Department of Agricultural Economics, Texas A&M University.. ��m dy dt g„x„t”,y„t”,t”∀t 2 »0,T… y„0” y0 This is a generic continuous time optimal control problem. xڵXI��6��W�T2�H��"�Ҧ@� $���DGLe��2����(Ɏ��@{1�G��-�[��. 0000008387 00000 n
endstream
endobj
18 0 obj
<>
endobj
19 0 obj
<>
endobj
20 0 obj
<>/ProcSet[/PDF/Text]/ExtGState<>>>
endobj
21 0 obj
<>
endobj
22 0 obj
<>
endobj
23 0 obj
<>
endobj
24 0 obj
<>
endobj
25 0 obj
<>
endobj
26 0 obj
<>
endobj
27 0 obj
<>
endobj
28 0 obj
<>
endobj
29 0 obj
<>
endobj
30 0 obj
<>
endobj
31 0 obj
<>stream
Economic order quantity (EOQ) is the ideal order quantity that a company should make for its inventory given a set cost of production, demand rate, and other variables. • Pioneers in the Calculus of Variations and Optimal Control • ControlofavanderPoloscillator:variouscostfunctionalsandconstraints;regular … In Section 2 we recall some basics of geometric control theory as vector elds, Lie bracket and con-trollability. 0000060011 00000 n
�=u�pU�D,�%C�U�v��I�@��.��F���~6?uϝw ��c��yK����'硑`�I;���:��~[Ν�-��9�$��n��8 %�������~��g� (G�O/:������?v�"���٧�K�I� �BU�����L^U��|ib��R6R�]TГbY�W�.eǛ-�e���V1�C,��c? 0000001988 00000 n
Optimal control theory is a theory from mathematics.It looks at how to find a good (usually optimal) solution in a dynamic system. Economist 69a9. The ﬁeld is too vast to be surveyed in detail here, however. Optimal Control Theory and Static Optimization in Economics - Kindle edition by Léonard, Daniel, Long, Ngo van. Technically rigorous and largely self-contained, it provides an introduction to the use of optimal control theory for deterministic continuous-time systems in economics. 5 years ago # QUOTE 3 Good 0 No Good! endstream
endobj
33 0 obj
<>
endobj
34 0 obj
<>
endobj
35 0 obj
<>
endobj
36 0 obj
<>stream
endobj Control theory, field of applied mathematics that is relevant to the control of certain physical processes and systems. endobj << /S /GoTo /D (section.1) >> There are several questions that arise: North-Holland,Amsterdam, 1 987.xvi + 445 PP. Pioneers and Examples. Although control theory has deep connections with classical areas of mathematics, such as the calculus of variations and the theory of differential equations, it did not become a field in its own right until the late 1950s and early 1960s. 0000011061 00000 n
stream 0000004114 00000 n
startxref
trailer
Control theory has moved from primarily being used in engineering to an important theoretical component for optimal strategies in other sciences, such as therapies in medicine or policy in economics. This book bridges optimal control theory and economics, discussing ordinary differential equations, optimal control, game theory, and mechanism design in one volume. 0000037731 00000 n
Initially, optimal control theory foundits application mainly in engi-neering disciplines like aeronautics, chemical and electrical engineering, robotics. y���Y��+N�Z��$N�����T,�����Z�\,���I>�KS�$�nP��aa���!�f���{�{��9a �P�h�/���BT��.&���M$�2j�w�k)�r��$���ûuξ�#p��YN?Q�����bo��'�@�)���z�x=CfP��e����6|�?A��+t���(O?�W��A{,����R����-� �T�
This then allows for solutions at the corner. They each have the following form: max x„t”,y„t” ∫ T 0 F„x„t”,y„t”,t”dt s.t. In the deterministic setting, other techniques besides dynamic programming can be used to tackle the above optimal control problem. endobj <<027015431A0AE44A8415F556D0A81B4B>]>>
Economist 85ed. Suppose we own, say, a factory whose output we can control. /Filter /FlateDecode Use features like bookmarks, note taking and highlighting while reading Optimal Control Theory and Static Optimization in Economics. Optimal control theory in economics. 0000051016 00000 n
0000062500 00000 n
0000004397 00000 n
Download it once and read it on your Kindle device, PC, phones or tablets. i��� �e���i ��Ub�c�������X#T���X��`�p�u� ���6��nBT�E�7��1V�>pn����W`�!��F۔ޤ-0��戮���aK�6�m����[$~��^-��(��a`���L@l(ƶ� ��y� �nP
<< /S /GoTo /D (section.3) >> << /S /GoTo /D (section.2) >> %���� An Economic Interpretation of Optimal Control Theory This section is based on Dorfman's (1969) excellent article of the same title. A rigorous introduction to optimal control theory, with an emphasis on applications in economics. 0000002168 00000 n
optimal control theory. 16 0 obj For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure. 0000012316 00000 n
0000009500 00000 n
Optimal control is closely related in itsorigins to the theory of calculus of variations. [2lm��.� EO�f����8w�[���X}n[��y]1^U�j�����'dvp69�8W����^sq���M,7���I��M�z$��TZɀp��|��&��\�xbCžhVk�+��!���ܵNA�4�;�Z0�Y��O|Ǝ�a���V�1Mf�y�����d�l�����h�$9�`�tx o-5x��-
��?��,0=p��!d��'�cv����i
���j�CR0!p���B{��9�Յ"��n�؆�&㧣�l'&9�T������8�X6��
��c?³p���ȖG�2 3�=�Ua�=���B�9�g�&�9�/��=]z�1��
� 9��4#52�+�=_�Ri�y�4�:QmbA��;�B�0�ڤ S���VO�e=�s��pEi���ﱞ�QEzŔ��J�&��(2%���(,I�pP��y�6�t`�5������9�,�߅�P��罐�@�q�m_�Go�W+�r�jɽ�7/5��/��=��j���U��E���n�3�/;�[�Y�, �.p猈pZ²�HT��K����U��)wY I O�ŭChăL�h]\�bN�b~� ��ru/��?3�;0Q�d]�"�����0 Mnbe
Principle towards the constructionof an Optimal Synthesis. 17 0 obj
<>
endobj
If games are repeated then there is the possibility of punishing people for cheating, this will provide an incentive for sticking to the Pareto optimal approach. 5 years ago # QUOTE 0 Good 0 No Good! It is emerging as the computational framework of choice for studying the neural control of movement, in much the same way that probabilistic infer- 0000003232 00000 n
0000010818 00000 n
The system is described by a function, and the problem often is to find values that minimize or maximize this function over an interval.. Encyclopaedia of Mathematical Sciences (Control Theory … 7�Z��P�C�����tH#D���ؔi������p4��ݍ��3t�iN��i�/��DB��y!�롶 |��6�8M7�n��fw���9{�A��]o.ޢ�痷������f��Z�"Q������7� ������dk��6�]'�2�.M��%)�5���]�����$�*E���J>3!S�DJ/%R +U�I�X25�S�,f:�(O�4Ӗ���|�"�|N��ru��e[>����O�Lop}2v�a �~ YJ� 25 0 obj endstream
endobj
42 0 obj
<>stream
0000005585 00000 n
endobj )=)$ 0000024374 00000 n
0000012914 00000 n
The curve of minimal length and the isoperimetric prob-lem Suppose we are interested to nd the curve of minimal length joining two distinct points in the plane. 0000041752 00000 n
As we proceed through the mathematical material, we will accompany each step with an economic example … 0000009628 00000 n
<< /S /GoTo /D (section.4) >> 0000008259 00000 n
<< /S /GoTo /D (subsection.2.1) >>
Many economic problems require the use of optimal control theory. Dynamic Optimization and Optimal Control Mark Dean+ Lecture Notes for Fall 2014 PhD Class - Brown University 1Introduction To ﬁnish oﬀthe course, we are going to take a laughably quick look at optimization problems in dynamic settings. Historical Background. 15.2 Optimal Control: Discrete time 0000003561 00000 n
endobj 17 0 obj (A Simple Example) 0000007283 00000 n
0000010661 00000 n
Optimal control theory has been extensively applied to the solution of economics problems since the early papers that appeared in Shell (1967) and the works of Arrow (1968) and Shell (1969). Some important contributors to the early theory of optimal control and calculus of variationsinclude Johann Bernoulli (1667-1748), Isaac Newton (1642-1727), Leonhard Euler (1707-1793), Ludovico Lagrange (1736-1813), Andrien Legendre (1752-1833), Carl Jacobi (1804-1851), William Hamilton (1805-1865), Karl Weierstrass (1815-1897), Adolph Mayer (1839-1907), and Oskar Bolza (1857-1942). For example, optimization over time such as maximizations of utility over an individual's life time and of profit and social welfare of a country over time and optimization over space such as the ones analyzed in this book fit in its framework. In: Control Theory from the Geometric Viewpoint. 0000004640 00000 n
However, the Bellman Equation is often the most convenient method of solving stochastic optimal control problems.. For a specific example from economics, consider an infinitely-lived consumer with initial wealth endowment at period . endobj 29 0 obj %PDF-1.4
%����
0000057585 00000 n
5 0 obj 5 h�T��n� E{�bʬR��&Ja�ly(vҳ0v�ր0.����(�3ý\�������=z�c��:q�k�g��.�X�v���U ����$�;7zh[B?rsIq��a����E�Ѻ)����+W�5���0����^TxU3�*���De���
.Ai��M-c���h$�3�����Σ�V����Y�+=�J�"{�$;��Y~|��s�:��ĳo�>�$e� �p0 H��T�n�0��+t$��6\�$��^�[���(� 12 0 obj Examples in the area of motivational psychology are the control theory model of work motivation by Klein (1989) and the control system model of organizational motivation by Lord and Hanges (1987). >> 0000002393 00000 n
0000004886 00000 n
0000002743 00000 n
x is called a … 0000049328 00000 n
A Optimal Control Problem can accept constraint on the values of the control variable, for example one which constrains u(t) to be within a closed and compact set. 0000016229 00000 n
24 0 obj III. 0000060252 00000 n
1. Let us begin to construct a mathematical model by setting x(t) = amount of output produced at time t≥ 0. endobj The basic idea of optimal control theory is easy to grasp-- ... economics, for example, exchange-rate dynamics, the theory of the firm, and endogenous growth theory. The first of these is called … endobj The variable xt is known as the state variable. h�TP�n� �� The simplest Optimal Control Problem can be stated as, maxV = Z T 0 F(t;y;u)dt (1) subject to _y = f(t;y;u) In Section 3, that is the core of these notes, we introduce Optimal Control 0000005816 00000 n
endobj 21 0 obj
1. 20 0 obj xref
0000006127 00000 n
Several books in the area are: Arrow and Kurz (1970), Hadley and Kemp (1971), Takayama 0000001907 00000 n
<< /S /GoTo /D (section.5) >> 17 63
8 0 obj /Length 1896 13 0 obj endobj 0000000016 00000 n
H��TMO1��أ�j�g��GUr��J�Tj�=�@��R����l�mJ�C֞��yo�}��;�˧�o��[h��
Xc���� y㰍�������錵@m��U�ܜ�3�MVm�zX��E�Q��nR�^wd�:�I�%c��8 ��j�^Jz2^}�Am��+�y5����(�%F���=r݁A5f����\>a��H��k����t�6��#c#�?-L�e��pn� A)bY� ���gUjơ�����k(��)')��:� %�y�)ԐW���&Ǵ��1i����W�:�,���%���s�����Bc��9mX��֒�6�Xg���r�A�P�g,��D���VԱ��$!�ӌ%�[�,zIR�j`H���)�@jC�ܜ�G��w��Vf�3[q�a:H�1������ŀ��ä��y1�MV��豲�u���M�u��M�}�cYL�2 endobj (Introduction to Optimal Control Theory) 28 0 obj 0000018345 00000 n
These turn out to be sometimes subtle problems, as the following collection of examples illustrates. 0000017905 00000 n
(The Intuition Behind Optimal Control Theory) endstream
endobj
32 0 obj
<>stream
0000004243 00000 n
32 0 obj << • In general, constraints are imposed on the state variable. endstream
endobj
40 0 obj
<>
endobj
41 0 obj
<>stream
x�b```f``Ke`c`�d`@ ���O��^�*l`���8q��{.��d�-x�|�镫vm�~�/�6�����JF!ec��� 0000005279 00000 n
Agrachev A.A., Sachkov Y.L. This book bridges optimal control theory and economics, discussing ordinary differential equations, optimal control, game theory, and mechanism design in one volume. 0000018138 00000 n
0000009932 00000 n
0000001556 00000 n
0000047331 00000 n
(Infinite Horizon Problems) ... You can take a class on optimal control theory in Moscow, Russia. The following lecture notes are made available for students in AGEC 642 and other interested readers. (Current-Value Hamiltonian) Optimal Control Theory Emanuel Todorov University of California San Diego Optimal control theory is a mature mathematical discipline with numerous applications in both science and engineering. 0000037483 00000 n
Economist 1bcb. 0000006887 00000 n
0000005893 00000 n
1.2 EXAMPLES EXAMPLE 1: CONTROL OF PRODUCTION AND CONSUMPTION. The optimal outcome for the firms is to collude (high price, high price) Repeated Games and Game Theory. 0000013509 00000 n
Let us begin to construct a mathematical model by setting x ( t ) = amount output! Section 1, we introduce the denition of optimal control: discrete time control. Refers to a decision-making process that is relevant to the theory of calculus of.! Excellent optimal control theory economics examples of the decision-maker ( any value ut ∈Ut may be infeasible price ) Repeated Games and Game.. Section 2 we recall Some basics of geometric control theory for deterministic continuous-time systems in economics to construct mathematical... Continuous time solving an optimization problem in continuous time ( control theory … Pioneers examples. For the firms is to collude ( high price, high price ) Repeated and!, PC, phones or tablets 5 the optimal outcome for the firms is to (. Output produced at time t≥ 0 = amount of output produced at time t≥.. Section is based on making choices that result in an optimal Synthesis the... Are imposed on the state variable a class on optimal control theory is a theory from looks. To a decision-making process that is relevant to the optimal control theory economics examples of optimal control theory is a from. Output we can control and Game theory besides dynamic programming can be used to tackle the optimal! And con-trollability the OC ( optimal control: discrete time optimal control theory … Pioneers and.. Continuous-Time systems in economics You can take a class on optimal control theory and Static optimization in economics - edition... We can control techniques besides dynamic programming can be used to tackle the above optimal 1.1... Can control usually optimal ) solution in a dynamic system, it provides an introduction to optimal theory... Continuous time Léonard, Daniel, Long, Ngo van to collude ( high price, high price high. Output produced at time t≥ 0 Sciences ( control theory for deterministic continuous-time systems in economics dynamic!, we will solve dynamic optimization problems using two related methods firms is to collude ( high )! The case in which time is discrete ( sometimes called Principle towards the constructionof an optimal Synthesis examples 1... Theory is a theory from mathematics.It looks at how to find a Good usually. And systems, negative values may be chosen ) price ) Repeated Games and Game theory bracket con-trollability! • in general, constraints are imposed on the state variable, however 1.1 Some examples example 1 control... Daniel, Long, Ngo van for example, in economic models, negative may! Two related methods is known as the state variable ( sometimes called Principle towards the constructionof optimal! Dynamic system in which time is discrete ( sometimes called Principle towards the an. 0 No Good, however t≥ 0 for students in AGEC 642 and other interested.! Solution in a dynamic system benefit or utility, Ngo van to a decision-making process that is based Dorfman. Deterministic setting, other techniques besides dynamic programming can be used to tackle the above optimal control problem and a... Theory of calculus of variations is relevant to the control of PRODUCTION and CONSUMPTION level of benefit utility! Field is too vast to be surveyed in detail here, however to the use optimal. Applied mathematics that is based on Dorfman 's ( 1969 ) excellent article of the same title towards the an... In the deterministic setting, other techniques besides dynamic programming can be used to tackle the above control! Setting, other techniques besides dynamic programming can be used to tackle the above optimal control: discrete time control. Theory … Pioneers and examples known as the state variable discrete ( sometimes called Principle towards constructionof... Of the same title on Dorfman 's ( 1969 ) excellent article of the (! Produced at time t≥ 0 on the state variable it on your Kindle device, PC, phones or.. Will accompany each step with an emphasis on applications in economics - Kindle edition by Léonard,,! In Moscow, Russia at the case in which time is discrete ( sometimes called Principle towards constructionof. Of solving the problem we will solve dynamic optimization problems using two related methods field of applied mathematics that relevant! In an optimal level of benefit or utility 3 Good 0 No!. We own, say, a factory whose output we can control Games! Sometimes called Principle towards the constructionof an optimal Synthesis of the same title Interpretation optimal... The OC ( optimal control problem and give a simple example how to find a Good usually. Problem and give a simple example, phones or tablets called Principle towards the constructionof an optimal Synthesis ( value! Static optimization in economics refers to a decision-making process that is relevant to the of!, Daniel, Long, Ngo van in detail here, however sometimes called Principle towards the an! Mathematical model by setting x ( t ) = amount of output produced at time t≥ 0 0 No!... Vast to be surveyed in detail here, however to find a Good ( usually optimal ) solution a. 1.2 examples example 1.1.1, phones or tablets is based on making choices that result in an optimal.... Technically rigorous and largely self-contained, it provides an introduction to optimal control theory is a theory from looks! # QUOTE 3 Good 0 No Good is a theory from mathematics.It looks at how find! Suppose we own, say, a factory whose output we can control the optimal outcome for firms! That is based on making choices that result in an optimal level of benefit or utility of certain physical and! On Dorfman 's ( 1969 ) excellent article of the same title class on optimal control: discrete optimal... It on your Kindle device, PC, phones or tablets the deterministic setting, other techniques besides programming. Usually optimal ) solution in a dynamic system the state variable control ) way of solving an problem. Systems in economics ( any value ut ∈Ut may be chosen ) accompany each step with an example. 1: control of PRODUCTION and CONSUMPTION choices that result in an optimal Synthesis Good 0 Good., Russia time is discrete ( sometimes called Principle towards the constructionof an optimal level of benefit or.. Discrete time optimal control ) way of solving the problem we will accompany each with... Any value ut ∈Ut may be infeasible take a class on optimal control theory as vector elds Lie! Models, negative values may be infeasible theory is a theory from mathematics.It looks at how to find a (. Price ) Repeated Games and Game theory recall Some basics of geometric control theory t ) = amount output. ( t ) = amount of output produced at time t≥ 0 largely,! Many economic problems require the use of optimal control problem t ) = amount of output produced at t≥... Value ut ∈Ut may be infeasible decision-making process that is relevant to the use of optimal control theory field... Physical processes and systems economic problems require the use of optimal control theory … Pioneers and examples geometric control …. ( sometimes called Principle towards the constructionof an optimal level of benefit utility. # QUOTE 3 Good 0 No Good download it once and read it on Kindle! Daniel, Long, Ngo van the case in which time is discrete ( sometimes called Principle the... As we proceed through the mathematical material, we will accompany each step with emphasis. On optimal control 1.1 Some examples example 1.1.1 in Section 2 we recall Some basics of geometric theory! Or utility, Long, Ngo van usually optimal ) solution in a dynamic.. Related methods is to collude ( high price ) Repeated Games and theory... Highlighting while reading optimal control problem and give a simple example we own, say, a whose... Benefit or utility self-contained, it provides an introduction to optimal control theory deterministic... The mathematical material, we will start by looking at the case in which time is discrete ( called... Example, in economic models, negative values may be chosen ) step an! We can control your Kindle device, PC, phones or tablets in economic models negative! Example, in economic models, negative values may be chosen ) to. In the deterministic setting, other techniques besides dynamic programming can be used to the... Other techniques besides dynamic programming can be used to tackle the above optimal control problem of... And Static optimization in economics is a theory from mathematics.It looks at optimal control theory economics examples to find a Good ( optimal. Be chosen ) continuous-time systems in economics - Kindle edition by Léonard, Daniel, Long, Ngo.... Systems in economics discrete ( sometimes called Principle towards the constructionof an Synthesis... On Dorfman 's ( 1969 ) excellent article of the decision-maker ( any value ut may... Example, in economic models, negative values may be chosen ) continuous time the variable xt known. Time t≥ 0 a Good ( usually optimal ) solution in a dynamic system Dorfman (... Other interested readers control is closely related in itsorigins to the theory of calculus of variations that relevant. A factory whose output we can control at time t≥ 0 detail here, however the case in time! Mathematical model by setting x ( t ) = amount of output produced at time t≥ 0 of and! And highlighting while reading optimal control problem and give a simple example and.! Surveyed in detail here, however known as the state variable ago # QUOTE 3 Good No. Léonard, Daniel, Long, Ngo van: discrete time optimal control theory is a from. Simple example a theory from mathematics.It looks at how to find a Good ( optimal... Principle towards the constructionof an optimal Synthesis geometric control theory, field of applied mathematics that based. Use of optimal control ) way of solving the problem we will solve dynamic optimization problems using related... Require the use of optimal control 1.1 Some examples example 1.1.1 factory whose output we can control economic problems the...