Opt imal Control of a Fini te D a m with Diffusion Input and

K e y w o r d s O p t i m a l control of dams, Diffusion processes, Potential, Infinitesimal generator, Control policies, Total discounted cost, Long-run average cost. 1. I N T R O D U C T I O N A N D S U M M A R Y A b d e l H a m e e d a n d N a k h i [1] d i scuss t h e o p t i m a l con t ro l of a f in i te d a m us ing P~,I r pol ic ies (def ined be low) . T h e y use b o t h t h e t o t a l d i s c o u n t e d cost as well as t h e l o n g r u n ave rage cost pe r a u n i t t i m e c r i t e r ion . T h e y a s s u m e t h a t t h e w a t e r i n p u t is a W i e n e r p roces s a n d W i e n e r p rocess re f l ec ted a t 0. A b d e l H a m e e d [2] t r e a t s t h e case of a n in f in i t e d a m w h e r e t h e i n p u t p rocess is a c o m p o u n d Po i s s on process . Bae , K i m a n d Lee [3] t r e a t t h e case of a f ini te d a m w i t h a c o m p o u n d P o i s s o n i n p u t ; t h e y o n l y d iscuss t h e l o n g r u n ave rage cost pe r a u n i t t i m e case. F a d d y [4] cons ide r s t h e case of a f in i te d a m w i t h W i e n e r i n p u t a n d p M policy. In all of t h e a b o v e pape r s , i t is a lways a s s u m e d t h a t t h e re lease r a t e s a re c o n s t a n t s a n d do not d e p e n d on t h e w a t e r c o n t e n t in t h e dam. In t h i s p a p e r , we cons ide r t h e case of a f in i te d a m w h e r e t he i n p u t p roces s is a d i f fus ion p rocess a n d t h e re lease r a t e s a re s t a t e d e p e n d e n t . Specifically, cons ide r a f in i te d a m w i t h c a p a c i t y v a n d a s s u m e t h a t t h e w a t e r i n p u t I = {I t ; t E T~+} is a 0 8 9 8 1 2 2 1 / 0 6 / $ see front matter (~) 2006 Elsevier Ltd. All rights reserved. dokl 0.1016/j.camwa.2005.11.006 Typeset by AA,~-'I~X 31.8 M . S . ABDEL-HAIvIEED AND Y. A. NAKHI diffusion process reflected at 0, v, and diffusion coefficients, (z) ~2 (z v) 3 . (v z) 2 .V2 + V (72 (Z) -(72 (V-Z)4 y2 where # is a real number and a 2 is a nonnegat ive number. I t follows t ha t I has s ta te space [0, v). Throughout , we will let 7~+ be the set of nonnegat ive real numbers. Suppose tha t a dam has capaci ty v. Let Z -(Zt; t 9 ~ + ) be the process describing the content of the dam. We restr ict ourselves to policies in which the release rate is zero until the water level reaches level ,\ (0 < A < v), when the water is released at ra te M ( z ) M ( v z ) 2, V until it reaches level r , (0 _< r < A). Once the level r is reached, the release ra te remains zero until the level A is reached again, and the cycle is repeated. I t is clear t ha t the content process is a delayed regenerat ive process with regenerat ion points being the t imes of successive visits to s ta te A. During a given cycle, the water content is a diffusion process reflected at v wi th diffusion coefficients, ~* (z) o2 (z v) a ~* (~ z) 2 V 2 + V where #* = # M and cr2(z), denoted by I* = {I~; t 9 ~ + } , and remains so until it drops to level r ; from then on and until it reaches A again the content of the dam behaves like a diffusion process wi th diffusion coefficients #(z) and o2(z). At any t ime the release ra te increases from 0 to M(z) a s ta r t ing cost K1M is incurred, and at any t ime the release ra te is decreased from M(z) to 0 a closing cost K2M is incurred. Moreover, for each unit of output , a reward A (which can be assumed to be 1) is received, and there is a penal ty cost which accrues at a ra te f , where f is a bounded measurable function. In Section 2, we describe the content process and give basic and main formulas for comput ing the cost functionals. In section 3, we give explicit expressions for the to ta l discounted cost over the infinite horizon as well the long-run average cost per a unit of t ime. 2 . B A S I C A N D M A I N R E S U L T S Throughout , we will let R = (Rt; t 9 7~+) and Z = (Zt; t E 7~+) denote the dam content and the release rates, respectively. The content process is best descr ibed by the bivar ia te process B = (Z, R), from the definition of the type of control policies deal t with, we have B0 = (0, 0). I t should be clear t ha t the process has s ta te space, E = ((l,X) • {0})U ([r, V) • { M } ) , where l denotes the lower bound of the s ta te space of I . The pena l ty cost occurs at a ra te given by f (Z, r) ---~" g ( z ) , L g* ( z ) , where g : (I,A) --+ ~ + and g* : [r ,Y] -+ ~ + following s topping times, (z , ~) 9 (/, ~) • { 0 } , ( z , r ) C [% V] x { M } , are bounded measurable functions. Define ti le T# = inf {t ETr : Zt = A}, T~ = i n f { t c T r : Z t = r } , Tn A = inf {t >_ TnA_I : Zt = A}, T,~ = i n f { t _ > T ~ : Z t = r } , n > l . (2.1)