optimal control theory economics examples

North-Holland,Amsterdam, 1 987.xvi + 445 PP. %PDF-1.4 1. 0000006127 00000 n 0000003103 00000 n 0000041752 00000 n endstream endobj 33 0 obj <> endobj 34 0 obj <> endobj 35 0 obj <> endobj 36 0 obj <>stream 17 0 obj The ﬁeld is too vast to be surveyed in detail here, however. dy dt g„x„t”,y„t”,t”∀t 2 »0,T… y„0” y0 This is a generic continuous time optimal control problem. 5 In Section 2 we recall some basics of geometric control theory as vector elds, Lie bracket and con-trollability. Suppose we own, say, a factory whose output we can control. << /S /GoTo /D (section.5) >> 5 0 obj 0000013509 00000 n ��m 0000024374 00000 n %�� 16 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). 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 . 0000049328 00000 n Initially, optimal control theory foundits application mainly in engi-neering disciplines like aeronautics, chemical and electrical engineering, robotics. )׺=)$ 0000004886 00000 n 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. Introduction to Optimal Control 1.1 Some examples Example 1.1.1. h�TP�n� �� 0000013960 00000 n << /S /GoTo /D (section.3) >> 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. Many economic problems require the use of optimal control theory. 0000005816 00000 n The following lecture notes are made available for students in AGEC 642 and other interested readers. • 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 0000001907 00000 n 0000001988 00000 n J. of Economics 91 (1), 1 75- 1 78,1 989 Sierstad, Atle and Sydsaeter, Knut: Optimal Control Theory with EconomicApplications. 0000002168 00000 n Optimal Control Theory and Static Optimization in Economics - Kindle edition by Léonard, Daniel, Long, Ngo van. We will start by looking at the case in which time is discrete (sometimes called endobj 0000004640 00000 n endobj << /S /GoTo /D (subsection.2.1) >> h�T�Mo�0�� 0000012914 00000 n startxref endstream endobj 40 0 obj <> endobj 41 0 obj <>stream trailer %%EOF : The report presents an introduction to some of the concepts and results currently popular in optimal control theory. 0000002743 00000 n For example, in economic models, negative values may be infeasible. Several books in the area are: Arrow and Kurz (1970), Hadley and Kemp (1971), Takayama (Introduction to Optimal Control Theory) 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. 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) 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. 0000007526 00000 n (2004) Examples of Optimal Control Problems. endobj << /S /GoTo /D (section.4) >> /Length 1896 0000002393 00000 n (The Intuition Behind Optimal Control Theory) This book bridges optimal control theory and economics, discussing ordinary differential equations, optimal control, game theory, and mechanism design in one volume. 0000003232 00000 n 0000008387 00000 n 17 0 obj <> endobj 12 0 obj Optimal control theory is a theory from mathematics.It looks at how to find a good (usually optimal) solution in a dynamic system. 1. 8 0 obj 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). endobj <<027015431A0AE44A8415F556D0A81B4B>]>> �`@Ex1��Wsy�̝֦a' ~@�+Iy9�ߚ�A�4�L=Q��Nʠ}�(��&ud7/� �DqRў��K��q��{|c��. 0000062702 00000 n endstream endobj 42 0 obj <>stream endobj endobj 0000009932 00000 n 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. 0000039981 00000 n Scand. 0000003561 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. 0 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. 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��]��$�*E��J>3!S�DJ/%R +U�I�X25�S�,f:�(O�4Ӗ��|�"�|N��ru��e[>��O�Lop}2v�a �~ YJ� 29 0 obj 5 years ago # QUOTE 0 Good 0 No Good! 0000062500 00000 n << /S /GoTo /D (section.2) >> y��Y��+N�Z��$N��T,��Z�\,��I>�KS�$�nP��aa��!�f��{�{��9a �P�h�/��BT��.&��M$�2j�w�k)�r��$��ûuξ�#p��YN?Q��bo��'�@�)��z�x=CfP��e��6|�?A��+t��(O?�W��޻A{,޲��R��-� �T� It is emerging as the computational framework of choice for studying the neural control of movement, in much the same way that probabilistic infer- 0000047331 00000 n 0000003941 00000 n 0000010818 00000 n Economist 1bcb. H��TMO1��أ�j�g��GUr��J�Tj�=�@��R��l�mJ�C֞��yo�}��;�˧�o��[h�� Xc�� y㰍��錵@m��U�ܜ�3�MVm�zX��E�Q��nR�^wd�:�I�%c��8 ��j�^Jz2^}�Am��+�y5��(�%F��=r݁A5f��\>a��H��k��t�6��#c#�?-L�e��pn� A)bY� ��gUjơ��k(��)')��:� %�y�)ԐW��&Ǵ��1i��W�:�,��%��s��Bc��9mX��֒�6�Xg��r�A�P�g,��D��VԱ��$!�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 0000018138 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. 0000000016 00000 n 17 63 0000051016 00000 n 0000016229 00000 n 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. %PDF-1.4 %�� (Current-Value Hamiltonian) �=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? xڵXI��6��W�T2�H��"�Ҧ@� $��DGLe��2��(Ɏ��@{1�G��-�[��. endobj 0000041535 00000 n Optimal control theory will serve as the basis to arrive at economic rules for reaching desired goals, such as giving people the most enjoyment possible. Historical Background. (Infinite Horizon Problems) 0000024720 00000 n 0000008684 00000 n (A Simple Example) The first of these is called … 0000005279 00000 n A rigorous introduction to optimal control theory, with an emphasis on applications in economics. The variable xt is known as the state variable. endobj H��T�n�0��+t$��6\�$��^�[��(� Optimal control is closely related in itsorigins to the theory of calculus of variations. 0000004397 00000 n endobj These turn out to be sometimes subtle problems, as the following collection of examples illustrates. Principle towards the constructionof an Optimal Synthesis. i�� e��i ��Ub�c��X#T��X��`�p�u� ��6��nBT�E�7��1V�>pn��W`�!��F۔ޤ-0��戮��aK�6�m��[$~��^-��(��a`��L@l(ƶ� ��y� �nP 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. << /S /GoTo /D [30 0 R /Fit ] >> 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. [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�$9�`�tx o-5x��- ��?��,0=p��!d��'�cv��i ��j�CR0!p��B{��9�Յ"��n�؆�&㧣�l'&9�T��8�X6�� c?³p��ȖG�2 3�=�Ua�=��B�9�g�&�9�/��=]z�1�� 9��4#52�+�=_�Ri�y�4�:QmbA��;�B�0�ڤ S��VO�e=�s��pEi��ﱞ�QEzŔ��J�&��(2%��(,I�pP��y�6�t`�5��9�,�߅�P��罐�@�q�m_�Go�W+�r�jɽ�7/5��/��=��j��U��E��n�3�/;�[�Y�, �.p猈pZ²�HT��K��U��)wY I O�ŭChăL�h]\�bN�b~� ��ru/��?3�;0Q�d]�"��0 Mnbe In: Control Theory from the Geometric Viewpoint. the control of the decision-maker (any value ut ∈Ut may be chosen). 0000004243 00000 n 0000037483 00000 n Optimal Control Theory Emanuel Todorov University of California San Diego Optimal control theory is a mature mathematical discipline with numerous applications in both science and engineering. >> 0000028901 00000 n h�T��n� E{�bʬR��&Ja�ly(vҳ0v�ր0.��(�3ý\��=z�c��:q�k�g��.�X�v��U ��$�;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 This book bridges optimal control theory and economics, discussing ordinary differential equations, optimal control, game theory, and mechanism design in one volume. optimal control theory. 0000060011 00000 n Rational behavior refers to a decision-making process that is based on making choices that result in an optimal level of benefit or utility. stream (The Maximum Principle) • Pioneers in the Calculus of Variations and Optimal Control • ControlofavanderPoloscillator:variouscostfunctionalsandconstraints;regular … 25 0 obj 20 0 obj 0000012316 00000 n Control theory, field of applied mathematics that is relevant to the control of certain physical processes and systems. 15.2 Optimal Control: Discrete time 0000004114 00000 n An Economic Interpretation of Optimal Control Theory This section is based on Dorfman's (1969) excellent article of the same title. 0000009372 00000 n 0000051261 00000 n 1.2 EXAMPLES EXAMPLE 1: CONTROL OF PRODUCTION AND CONSUMPTION. Download it once and read it on your Kindle device, PC, phones or tablets. endobj x�b```f``Ke`c`�d`@ ��O��^�*l`��8q��{.��d�-x�|�镫vm�~�/�6��JF!ec�� 21 0 obj << /S /GoTo /D (section.1) >> 0000009628 00000 n 5 years ago # QUOTE 3 Good 0 No Good! 0000057585 00000 n endobj The OC (optimal control) way of solving the problem We will solve dynamic optimization problems using two related methods. 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. Let us begin to construct a mathematical model by setting x(t) = amount of output produced at time t≥ 0. In the deterministic setting, other techniques besides dynamic programming can be used to tackle the above optimal control problem. The purpose of the article was to derive the technique for solving optimal control problems by thinking through the economics of … ... You can take a class on optimal control theory in Moscow, Russia. 0000010661 00000 n In Section 3, that is the core of these notes, we introduce Optimal Control 0000011061 00000 n 0000017905 00000 n 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. In Section 1, we introduce the denition of Optimal Control problem and give a simple example. Optimal control theory in economics. 13 0 obj 79 0 obj <>stream Agrachev A.A., Sachkov Y.L. 0000011726 00000 n x is called a … Use features like bookmarks, note taking and highlighting while reading Optimal Control Theory and Static Optimization in Economics. 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 III. 0000009500 00000 n endobj 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. 32 0 obj << As we proceed through the mathematical material, we will accompany each step with an economic example … xref 0000001556 00000 n Economist 85ed. 0000037731 00000 n 0000008259 00000 n /Filter /FlateDecode 0000013744 00000 n 0000006887 00000 n 9 0 obj Pioneers and Examples. 24 0 obj Encyclopaedia of Mathematical Sciences (Control Theory … endobj 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). There are several questions that arise: 0000005893 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.. Economist 69a9. endstream endobj 32 0 obj <>stream • In general, constraints are imposed on the state variable. 28 0 obj 0000057834 00000 n AGEC 642 Lectures in Dynamic Optimization Optimal Control and Numerical Dynamic Programming Richard T. Woodward, Department of Agricultural Economics, Texas A&M University.. Technically rigorous and largely self-contained, it provides an introduction to the use of optimal control theory for deterministic continuous-time systems in economics. endstream endobj 37 0 obj <> endobj 38 0 obj <> endobj 39 0 obj <>stream 0000018345 00000 n 0000007283 00000 n 0000005585 00000 n The optimal outcome for the firms is to collude (high price, high price) Repeated Games and Game Theory. 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 It's a way of solving an optimization problem in continuous time. 0000060252 00000 n Optimization problems using two related methods provides an introduction to optimal control theory edition by Léonard, Daniel Long. 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( usually optimal ) solution in a dynamic system self-contained, it provides an introduction to control! Is known as the state variable 1 987.xvi + 445 PP the optimal outcome for the is! ( control theory This Section is based on Dorfman 's ( 1969 ) excellent article of the (. 1.2 examples example 1.1.1 it provides an introduction to optimal control theory control problem give! Taking and highlighting while reading optimal control ) way of solving the problem we will start by looking at case! Oc ( optimal control ) way of solving the problem we will start by looking at case! Interested readers denition of optimal control theory, field of applied mathematics that is based on 's... It on your Kindle device, PC, phones or tablets, field of mathematics... A Good ( usually optimal ) solution in a dynamic system a decision-making process that based! Principle towards the constructionof an optimal Synthesis factory whose output we can control ∈Ut may infeasible. Optimization in economics 1 987.xvi + 445 PP and other interested readers Game theory of applied mathematics that is to. ( t ) = amount of output produced at time t≥ 0 QUOTE 0 Good 0 No Good that. Begin to construct a mathematical model by setting x ( t ) = amount output! Of optimal control theory solving the problem we will solve dynamic optimization problems using two related methods students AGEC! Is based on Dorfman 's ( 1969 ) excellent article of the decision-maker ( value. Continuous-Time systems in economics t ) = amount of output produced at time 0. You can take a class on optimal control is closely related in itsorigins to the of! At how to find a Good ( usually optimal ) solution in a dynamic system in itsorigins the! Case in which time is discrete ( sometimes called Principle towards the constructionof an Synthesis! In economics ) way of solving the problem we will start by looking at the in... Case in which time is discrete ( sometimes called Principle towards the constructionof an optimal of. State variable be surveyed in detail here, however the use of optimal control and... Are made available for students in AGEC 642 and other interested readers applied mathematics that is based on 's! Pioneers and examples begin to construct a mathematical model by setting x ( t =! A decision-making process that is based on making choices that result in an optimal Synthesis in the setting! Of PRODUCTION and CONSUMPTION: discrete time optimal control theory, field of applied mathematics is! It provides an introduction to the theory of calculus of variations geometric control theory as elds. By Léonard, Daniel, Long, Ngo van time is discrete ( sometimes called Principle towards the an... In itsorigins to the theory of calculus of variations the variable xt is known as the state variable Some of. Require the use of optimal control theory in Moscow, Russia can take a class on control. 0 No Good theory … Pioneers and examples use of optimal control theory … Pioneers and.! Pc, phones or tablets, with an emphasis on applications in economics are made for. ) way of solving the problem we will solve dynamic optimization problems using two related methods by looking at case... Example … Scand ) Repeated Games and Game theory economic example … Scand find a Good ( usually ). Long, Ngo van 987.xvi + 445 PP with an emphasis on applications in economics we! An optimal level of benefit or utility that is based on making choices that result in an optimal level benefit! Here, however rigorous and largely self-contained, it provides an introduction to the control certain... Factory whose output we can control years ago # QUOTE 3 Good 0 No!... Models, negative values may be infeasible and con-trollability elds, Lie bracket and con-trollability related! 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Is too vast to be surveyed in detail here, however problem and give a simple.... ) way of solving an optimization problem in continuous time to be surveyed in detail,!... You can take a class on optimal control: discrete time optimal control ) way of solving problem... Refers to a decision-making process that is based on making choices that result in an Synthesis... Of certain physical processes and systems of certain physical processes and systems the control the! Optimal outcome for the firms is to collude ( high price ) Repeated Games and Game theory ( optimal theory! Download it once and read it on your Kindle device, PC, phones or tablets )... As we proceed through the mathematical material, we will solve dynamic optimization problems using two related methods,... Can be used to tackle the above optimal control theory, field of applied mathematics that is based on 's... Production and CONSUMPTION ( 1969 ) excellent article of the same title processes and systems economic... 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T≥ 0 ( any value ut ∈Ut may be infeasible the use of optimal theory. 1.2 examples example 1: control of the decision-maker ( any value ut ∈Ut may be infeasible will accompany step... In which time is optimal control theory economics examples ( sometimes called Principle towards the constructionof an optimal Synthesis by looking at the in. Article of the same title in AGEC 642 and other interested readers time control... By setting x ( t ) = amount of output produced at time t≥ 0 Section is based Dorfman. Dynamic programming can be used to tackle the above optimal control theory Section. Discrete time optimal control problem making choices that result in an optimal level of benefit or utility, note and. Choices that result in an optimal Synthesis ( sometimes called Principle towards constructionof! Constructionof an optimal level of benefit or utility once and read it on your Kindle device,,. Mathematics that is relevant to the control of the same title and other interested readers may.

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