(Introduction to Optimal Control Theory) endobj 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���]�����$�*E���J>3!S�DJ/%R +U�I�X25�S�,f:�(O�4Ӗ���|�"�|N��ru��e[>����O�Lop}2v�a �~ YJ� The OC (optimal control) way of solving the problem We will solve dynamic optimization problems using two related methods. They each have the following form: max x„t”,y„t” ∫ T 0 F„x„t”,y„t”,t”dt s.t. These turn out to be sometimes subtle problems, as the following collection of examples illustrates. 0000060011 00000 n << /S /GoTo /D (section.5) >> 0000006127 00000 n endstream endobj 18 0 obj <> endobj 19 0 obj <> endobj 20 0 obj <>/ProcSet[/PDF/Text]/ExtGState<>>> endobj 21 0 obj <> endobj 22 0 obj <> endobj 23 0 obj <> endobj 24 0 obj <> endobj 25 0 obj <> endobj 26 0 obj <> endobj 27 0 obj <> endobj 28 0 obj <> endobj 29 0 obj <> endobj 30 0 obj <> endobj 31 0 obj <>stream 0000001907 00000 n 0000057834 00000 n 1. endobj 29 0 obj 0000004114 00000 n Several books in the area are: Arrow and Kurz (1970), Hadley and Kemp (1971), Takayama 0000057585 00000 n 24 0 obj 0000005279 00000 n Agrachev A.A., Sachkov Y.L. 0000001556 00000 n 0000013744 00000 n endstream endobj 40 0 obj <> endobj 41 0 obj <>stream The first of these is called … 0000037731 00000 n This book bridges optimal control theory and economics, discussing ordinary differential equations, optimal control, game theory, and mechanism design in one volume. Economist 85ed. Pioneers and Examples. x�b```f``Ke`c`�d`@ ���O��^�*l`���8q��{.��d�-x�|�镫vm�~�/�6�����JF!ec��� 5 0 obj endobj 0000002168 00000 n 0000051261 00000 n h�T�Mo�0��� [2lm��.� EO�f����8w�[���X}n[��y]1^U�j�����'dvp69�8W����^sq���M,7���I��M�z$��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 1.2 EXAMPLES EXAMPLE 1: CONTROL OF PRODUCTION AND CONSUMPTION. Optimal Control Theory and Static Optimization in Economics - Kindle edition by Léonard, Daniel, Long, Ngo van. %%EOF 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. The optimal outcome for the firms is to collude (high price, high price) Repeated Games and Game Theory. 0000013509 00000 n 0000041752 00000 n 13 0 obj 0000004640 00000 n 9 0 obj endstream endobj 32 0 obj <>stream The following lecture notes are made available for students in AGEC 642 and other interested readers. 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). Technically rigorous and largely self-contained, it provides an introduction to the use of optimal control theory for deterministic continuous-time systems in economics. 0000024720 00000 n 16 0 obj 0000009628 00000 n The variable xt is known as the state variable. 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. 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. 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 Rational behavior refers to a decision-making process that is based on making choices that result in an optimal level of benefit or utility. 0000003941 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 Optimal control theory is a theory from mathematics.It looks at how to find a good (usually optimal) solution in a dynamic system. xڵXI��6��W�T2�H��"�Ҧ@� $���DGLe��2����(Ɏ��@{1�G��-�[��. 0000060252 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.. 0000006887 00000 n 0000017905 00000 n Use features like bookmarks, note taking and highlighting while reading Optimal Control Theory and Static Optimization in Economics. 0000062702 00000 n 12 0 obj Initially, optimal control theory foundits application mainly in engi-neering disciplines like aeronautics, chemical and electrical engineering, robotics. Suppose we own, say, a factory whose output we can control. (A Simple Example) This then allows for solutions at the corner. (Current-Value Hamiltonian) 0000005893 00000 n 0000010818 00000 n 0000062258 00000 n J. of Economics 91 (1), 1 75- 1 78,1 989 Sierstad, Atle and Sydsaeter, Knut: Optimal Control Theory with EconomicApplications. dy dt g„x„t”,y„t”,t”∀t 2 »0,T… y„0” y0 This is a generic continuous time optimal control problem. (Infinite Horizon Problems) : The report presents an introduction to some of the concepts and results currently popular in optimal control theory. 0000024374 00000 n 0000007283 00000 n stream Control theory, field of applied mathematics that is relevant to the control of certain physical processes and systems. (The Intuition Behind Optimal Control Theory) 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). endstream endobj 42 0 obj <>stream Historical Background. Download it once and read it on your Kindle device, PC, phones or tablets. 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). We will start by looking at the case in which time is discrete (sometimes called endobj Encyclopaedia of Mathematical Sciences (Control Theory … 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. 0000041535 00000 n 0000002393 00000 n 0000012914 00000 n %PDF-1.4 %���� Economist 1bcb. 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 /Length 1896 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 . Start by looking at the case in which time is discrete ( sometimes Principle... Can be used to tackle the above optimal control theory, field of mathematics... 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