# The problem considered is that of approximate minimisation of the Bolza problem of optimal control

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(2) . where the functions. . Ê Ê . . in the problem, in Ê .. Ê Ê Ê , Ê Ê are given. . . . Ê, Ê is an open set, and . . .

(3) is a point. We additionally assume that the functions and. satisfy. . (Z). (b) . the Lipschitz condition with respect to the compound variable satisfies locally the Lipschitz condition in Ê . . Definition 1. A pair of functions is called admissible when it satisfies (2), (3) and the function is integrable. The trajectory is then called an admissible trajectory, and the control is then called an admissible control. Now we can restate the Bolza problem:. Find. , where the infimum is taken over all admissible pairs satisfying (4).. . . . Definition 2. Any admissible control, for which the minimum of the functional is reached, is called optimal control and denoted by .. . . An exact solution to the Bolza problem is rather hard to find. However, we can consider the problem of finding an approximate solution, i.e., such admissible pairs , defined on , for which

(4) , and the following condition holds:. . . . . . . . (5). . where is some real number. Every control obtained in this way is called an -optimal control. In this paper we are going to approximate the Bolza problem with methods of dynamic programming. Such.

(5) J. Pustelnik. 178 an approach is extensively described in the existing literature (Cesari, 1983; Fleming and Rishel, 1975). However, unlike most currently developed methods for solving Hamilton-Jacobi equations (and thus the Bolza problem), which combine the direct use of the classical approaches with clever numerical approximations, we use a distinct method developed in (Jacewicz, 2001). Recent publications in the field, which rely on the classical approach with innovative numerical solutions, include (Karlsen and Risebro, 2002), where the authors use a front tracking method developed for dealing with hyperbolic conservation laws to tackle Hamilton-Jacobi equations; (Kurganov and Tadmor, 2000), where another method originally developed for hyperbolic conservation laws is used, namely, the semi-discrete central schemes; (Bryson and Levy, 2001), where also central schemes are used; (Tang et al., 2003), where an adaptive mesh instead of a rectangular one is used for solving the H-J equation numerically. All those papers are focused on obtaining practical numerical solutions to the H-J equation, while other considerations, such as convergence or stability, are secondary. Among the few papers which approach the subject in a different manner there is the article (Szpiro and Dupuis, 2002), where the main emphasis is on the proof of the convergence of the proposed method. Szpiro and Dupuis develop a novel, probabilistic approach to H-J equations. However, the method proposed there is rather complicated. Our method is tailored to the Bolza problem and, in our belief, it is much simpler. It also has an interesting property—the result is being calculated with precision given a priori. The approximation is stable and convergent. Two most important contributions of this paper are the elimination of some artificial conditions that impose severe limitations on the acceptable class of functions which are stated in (Jacewicz, 2001) (Lemma 5) at the stage of constructing the approximate value function and an effective, easily adaptable to machine implementation, algorithm for constructing such approximated value functions. These limitations, expressed in the assumptions (L1)–(L3) of Lemma 5 (p. 415), have the following implications:. . . real-world problem that would fit into such limitations. Additionaly, the proofs of theorems and numerical calculations are also complicated, the latter because the choice of starting functions is limited by the assumption (L1). The current paper does not necessitate all those assumptions. It also simplifies the notation which makes the numerical algorithm shorter and easier to apply. However, in cases where Jacewicz’s method is applicable, the method presented here gives substantially the same results. Therefore, we did not find it necessary to present a comparison of those two methods on the same example. The paper is further structured as follows: Section 2 ends with the fundamental theorem called the verification theorem. It specifies conditions which have to be fullfilled by a function in order to be an -value function. In Section 3 we specify a method for constructing a function that is a suitable candidate to be an -value function. Section 4 shows how to calculate the values of an -value function at specified points. It also includes an example.. 2. Definition and Properties of a Value Function and an Approximate Value Function 2.1. Dynamic Programming—the Value Function. . Ê be a set with non-empty interior, Let covered by graphs of admissible trajectories, i.e., for every there exists an admissible pair , defined on such that and for . The assumption that the interior of is non-empty is essential for further deliberations and constitutes some limitation of this method.. . Definition 3. A function is called a value function when. . . . . defined in . . . . . . (L1) requires that the first derivative of the value. (6) where the infimum is taken over all admissible trajectories , which start from and their graphs are contained in .. (L2) requires that the graph of the function . 2.2. Dynamic Programming—the Approximate Value Function. function with respect to be non-zero inside any of the sets into which the domain is partitioned; . must lie between the graphs of two functions linear with respect to

(6). (L3) has some consequences limiting the allowed class of functions . . Because of those limitations, Jacewicz’s method can be considered only theoretical. It is very hard to find a. . . We will now discuss the approximation of the value function by an -value function. However, because we modify the definition of the -value function (compared with (Jacewicz, 2001)) and of the -optimal trajectory, the proof of the verification theorem (Thm. 1) will also be presented..

(7) A method for constructing -value functions for the Bolza problem of optimal control defined on Definition 4. A function the set is called an -value function iff . . . . . . . . . . . . . (7). . Definition 5. An admissible trajectory is called -optimal if for all admissible we have trajectories . . . . . .

(8). . . . . . . . (9). satisfying the boundary condition . Moreover, if is an admissible pair defined on , , such that for almost every we have. . . . . . . . . . . . . . . . . . . . . .

(9). (11). . . . . . . . (12) Applying the boundary condition and the definition of the value function, we get . . . . . (13). . . From (10) we have that along the , starting from , the following holds: . (14). for all trajectory . . . . . . . . . . . . . . . . . . From (14) and the boundary condition we obtain. . . . . . . . . . . . . . (15). which gives . . . . . . . . . . . This, combined with (13), proves that optimal trajectory and that . (16). . . is an -. . (10). then is an -optimal trajectory. Additionally, if there exists an admissible for some point trajectory starting from and satysfying (10), then where is some -value function.. . . . (8). . . . . . Theorem 1. Let Ê be an open set, be a value function in and be a function defined on , which is almost everywhere on a solution of the following inequality:. . . . Because (9) holds almost everywhere in , for every admissible trajectory starting from we obtain. An -optimal trajectory always exists because every set of real numbers has an infimum. We are now ready to formulate the fundamental theorem of the introductory part:. . . . . . is an admissible trajectory. Ob-. . . . . . . . . Obviously, for a given , the above -value function is not uniquely defined, and therefore we speak of many -value functions. One should notice that is finite in .. . . . where is a value function, is a function described in the formulation (1)–(4) of the Bolza problem that satisfies (Z), and is a fixed number, which will be assumed to be constant in all further deliberations.. . Proof. Assume that viously, we have. 179. 3. Construction of -Value Functions and a Computer Algorithm The method leading to the construction of -value functions was first described in (Jacewicz, 2001). However, our method does not need additional assumptions and is better suited to being used as a base for computer programs. In particulars, we do not assume anything about.

(10) J. Pustelnik. 180. and the functions that goes beyond assumptions listed in (Z) in the formulation of the Bolza problem and those additional assumptions that are the consequence of the verification theorem.. 3.1. Construction of -Value Functions. . . Let Ê be a compact set with a non-empty interior covered with graphs of admissible trajectories and let Ê be a compact set. We will begin the construction of the -value function by choosing some arbitrary function that satisfies the boundary condition .. . We will define on a function that will correspond to the right-hand side of the Hamilton-Jacobi equation:. . . . . . . . . (17) . . . . . . for all . . (18). defined above has in values of different signs and therefore it cannot satisfy (9). In order to find a function that satisfies the assumptions of the verification theorem, we will now define a family of , Æ These functions functions will satisfy for all the inequality (9), where Æ are numbers that depend on the chosen such that for . The function for every is described by the following formula and the construction Æ , is described below: of . . . . . . . . . .

(11). (19). . . . . . . . . . . . . . . . (23). A simple calculation yields. which means. . . . . . . . . . . . .

(12). . . . . Since is a compact set, the function reaches its bounds on , which we denote by and , respectively, . . . The function is continuous on . Moreover, it satisfies the Lipschitz condition on .. The function . . .

(13). and where the functions satisfy the conditions (Z), the function is defined as above and the infimum is replaced with the minimum due to the compactness of .. . . . . (20) (21) The sets constitute a covering of the set , i.e., for every and . We will now define the auxiliary functions and on the sets , as follows: (22). . We will begin the construction of for every by defining its domain. Let us divide the inÊ being the image of the set in terval the mapping creating a subinterval

(14)

(15) such that and that for all we have

(16)

(17) . Write Now we divide the set into the following subsets :. . . . . . (24). . (25) It is easy to notice that for some fixed we can always choose such that for every we have .. . We define the function . as follows:. . for . . (for fixed ) in . . . (26) the function . . . . Obviously, for every satisfies the inequality (9) of the verification theorem (Thm. 1) of the dynamic programming for some fixed , and satisfies the boundary condition of this theorem (since its values for are equal to the corresponding values of ), yet it is not a function of the. .

(18) A method for constructing -value functions for the Bolza problem of optimal control class (probably it is even a discontinuous function), and thus it does not fulfil the requirements. In order to satisfy the assumptions of the verification theorem, we have to smooth the function by convoluting it with a function of the class Ê having compact support. From now on we assume that (the number of sets ) is a fixed natural number, and ! is some real number. The function " Ê Ê Ê of the class Ê having compact support, where ! Ê , is defined as follows: Let " Ê Ê Ê be a function of Ê having a compact support, such that class # Ê , Ê " and supp " where ‘supp’ denotes the support, and # Ê for any $ Ê is a ball in Ê with the center at having the radius $ . Obviously, " ! " ! ! . It is easy to see that such a function " is an infinitely smooth function having the compact support supp " # Ê and Ê " Ê " An example of such a function will be given in the section devoted to the numerical algorithm. Let us now define for each ! Ê a new function :. . . . . . . . . . . . . ". . (27). . where the star denotes convolution. From a theorem in (Adams, 1975) we have that for every and ! the function is of the class , which means that the corresponding function , defined by. . . . . . . . Case 1:. . is continuous in . We will now try to evaluate the function Let % be a set where is discontinuous. The Lebesgue measure of % in Ê is zero, which is a consequence of the definition of .. . . . . Lemma 1. For every given, fixed and for every Æ there exists real !

(19) such that for every ! !

(20) and for all % the following inequality is satisfied:. . . . (29). Ê. . . . . Ê. . . . . . Case 2:. . . . . . . . . . . We will give here the proof only for Case 1, since the proof for Case 2 is analogous. Define. # # for every & &. & . & . Obviously, #. Ê . . . . . ! ! we have and & & . Ê Ê . . . . . . . . . . . . Thus. . . . . . " . Ê. . . . . . ". . . . (28). . . . . . . . . . . . Proof. Take the arbitrary %. Then for some. the point . Since is uniformly continuous on , we can always find ! such that for all # Ê we have the estimate Therefore, we have one of the following cases, in accordance with the location. of in the set :. . . 181. . . ". . . . . . ". . . . . . . . . . . . . . . " . . .

(21) J. Pustelnik. 182. . " . Ê. . . Ê. " . . Ê . . . . . (. . '. . . . . (32). . Ê. . ) . ). . . ' ( . !. . . . ) . (37). (38). . ). . ) . . . . . . . (33). ) *. From Lemma 2 we have that for every Æ there exists !

(22) such that for every ! !

(23) and for all . . % . Proof. We must show that for every real * there exists real ! such that for every ! ! and for all % we have. where ' is a Lipschitz constant of the function of the class . . . . . ). . ) . Ê . . . Lemma 3. For fixed and for all we have. " . Ê. . . (36). . . . . . . " . (35). and the convergence is uniform.. . Proof. We have the following estimate:. . % we have. In order to simplify the notation, we will define two auxiliary functions on ( % ! :. (31). Lemma 2. For a given, fixed and for every Æ there exists !

(24) such that for every ! !

(25) and for all % the following inequality holds:. (34). . . . . %. !

(26). . . (obviously, by taking !

(27) such that for every ! !

(28) we have ' ( !

(29) or, more simply, !

(30) from (30) we see that the inequal

(31) ity (31) holds).. . . and the convergence is uniform.. . . . There exists real !

(32) ! such that for every ! !

(33) we have. . Remark 1. For fixed and for all . where ' is a Lipschitz constant of a function of the class .. such that for every. . . (30). ! . . such that for every It is enough to choose ! !

(34) we have ' ( !

(35) or, more simply, . !

(36)

(37) . . . . . . . So there exists real !

(38) ! !

(39) we have. . . . . . . (39). ' . where ' is a constant limiting the function from above on % . Taking ! !

(40) where * we complete the proof. Æ is such that

(41) '. . .

(42) A method for constructing -value functions for the Bolza problem of optimal control Let us introduce some additional symbols:. . . % one of these cases holds, which For all proves the theorem.. . + ) ) . . +. . ). . . . ). . . . . (40). . where and are the values of control that minimize the respective functions ) and ) at the point . Lemma 4. For fixed and for all . . +. . % we have. + . (41). and this convergence is uniform. Proof. In order to prove the uniform convergence, we have to show that for an arbitrarily chosen real number * and for there exists Æ such that for every ! Æ all % the following inequality holds:. . + . + *. (42). % +

(43) + and , ¼ ¼¼% + ¼ + ¼¼ Obviously, we have , , % and , , ¼ We partition the set % into two sets: , ¼¼ . . . . . . . . . . . . . . ) . ) *. . (43). . The existence of such a number Æ is guaranteed by Lemma 4. Now we have two separate cases: ¼ Case 1: , . We have the following inequality:. . + . + + . + . . ) ) ) ) ) * (44) ¼¼ Case 2: , . In this case we have following inequality: + + + + ) ) ) ) ) ) * . ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (45).

(44). + . + . . . (46). Proof. It is obvious because of the uniform convergence of to + with respect to ! on % .. . +. Some readers may have noticed that !

(45) has not been computed effectively. We will give now a precise formula for !

(46) : in Lemma 3 we asserted that ! Because * we have from Lemma 4

(47) . . that. '. and therefore. . .

(48). which is equivalent to ' ,. !

(49) !

(50) . . ..

(51) . We are now ready to give the most important theorem of this paragraph.. . Theorem 2. For given, fixed and for any Æ there exists real !

(52) such that for every ! !

(53) and for all % the following inequality holds:. . . . . . . . (47). Proof. It results immediately from the following estimate, where ! !

(54) !

(55) !

(56) :. . + + . + + . . . . . . . . . (48). . . . Æ there exists ! , and for. Remark 2. For fixed and for any real !

(57) such that for every ! % we have all . Let * be some fixed real number and let Æ be a real number such that for every ! Æ and for % : all . . 183. As we can see, for every . % all . Æ and all ! !

(58) the values of the function can be estimated as follows:. . . is of the class. . . . Obviously, the function , yet it is not an -value function because does not take non-positive values close to zero on ..

(59) J. Pustelnik. 184 Of course, there is also one more important condition for our function to be an -value function, namely, the boundary condition. We will now give an estimate on how the values of differ from the values of which in turn are equal to the values of the original function .. . . . . . . . . . . . . (49). . . %. Then for some Proof. Take arbitrary . the point . Since is uniformly continuous on , we can always find ! such that for all # Ê we have the estimate Therefore, we have one of following cases, according to the location of . in the set :. . . . Case 1: Case 2:. . Ê Ê. . . . . . . . . . . . . . . . . . . We will give here a proof only for Case 1, since the proof for Case 2 is analogous. Let us introduce the following symbols: & . # #. . . Ob ! ! we have viously, for every # Ê & & and & & . . . . . . Ê Ê . . . . . . . . &. . . . . Thus the following inequality holds:. " Ê. " . " . . . . . . . ". . . . . . . . Ê. " . . . . . . ' ( ! . . . . . . . Ê. . . . . . (50). where ' is a Lipschitz constant of a function the class .. . of.

(60) ! such that for every There exists !

(61) we have !. . !. . . . . . . . . . (51)

(62) ! such that for every (obviously, by taking !

(63) we have ' ! ! ( !

(64) or, more

(65) , we see that the inequality simply, !

(66) (53) holds).. . Obviously, on the boundary (for ) we have the following estimate for all Æ and for every !.

(67) !. . . . . (52). We will now construct a function that will be a function of the class , and the respective function

(68) will take non-positive values close to zero almost everywhere on . Let us introduce the following definitions: .

(69). . . . . .

(70). . . . . . .

(71) . .

(72). . (53).

(73) . . . Ê. . . . . .

(74). . " . ". . . . . . . . Lemma 5. For every given, fixed and for every Æ there exists !

(75) such that for every !

(76) and for all ! % the following inequality is satisfied:. . . . . . . . (54). . Obviously, the function

(77) is of the class . At the same time, by simple calculations we have that for all ! !

(78) : . .

(79). . . . . (55).

(80) A method for constructing -value functions for the Bolza problem of optimal control so the following estimation holds:.

(81) . . . . (56). In addition to that we have the following inequality for the

(82) : boundary condition, which holds for all ! ! .

(83). . . . . . which in turn means that for all , where is such that it also holds . Therefore, the function

(84) ,

(85) satisfies (9) for all ! !

(86) ! and is a candidate for an -value function according to Theorem 1. By simple calculations we obtain that

(87) are easy to . . The values of !

(88) and ! calculate from the corresponding Lipschitz constants.. . . 3.2. Algorithm for Evaluating -Value Functions The algorithm for calculating the -value function is quite straightforward. However, it is a numerical algorithm, and not a computer algorithm. We shall discuss improvements needed for converting it into a computer program. Such an implementation is in fact being written as part of our research and will soon be submitted for publication. The main aim of this section is to provide a kind of summary for the method presented above, rather than a computeroriented algorithm. Therefore we require the user to submit data, which in a fully developed algorithm would be calculated by the program. Likewise, the algorithm offers no assistance in establishing the set . Input: The algorithm requires the following values to be calculated by the user: all required Lipschitz constants i.e., ' , ' ' and ' the values of lower and upper limits of and , the value of the point for which the -value function should be calculated and, of course, the function of the class satisfying the boundary condition. Without the boundary condition being satisfied the results will usually be wrong. The aforementioned values have to be provided by the user. However, in a computer implementation we will calculate suitable approximations.. . Output: The output of the algorithm is the value of the -value function at a given point The algorithm calculates the value of an -value function at a single point provided by the user. The algorithm consists of the following steps:. . Step 1: Calculate the values and and for all following steps set and . 185. Step 2: Compute the value ! , such that for all ! ! the function

(89) is an -value function, - Ê . based on the re-. . . . spective Lipschitz constants as follows:. . . !. where we assume so that !. . .

(90) !

(91) !

(92) !

(93) !

(94) !. . . . ' ( . . . . ' ' ( . (. . . . '. Step 3: For ! follows: " . . . !. . . . !. . . define the function " as. . . . . for . ! !. for. . Determine a constant - such that we shall have " In fact, we simply set: Ê Step 4: Determine. . . . . . . . . . . . .

(95). . . where denotes the integer part of , i.e., the greatest integer number that is less than . (We identify the right for whose we have where are defined in accordance with (20) and (21).). . . . Step 5: Compute . according to (22).. . . Step 6: Compute . . . Ê. . ". . . according to (27). Step 7: Determine .

(96). . . . . . according to (55). The value calculated in Step 7 is the output of the algorithm..

(97) J. Pustelnik. 186. References Adams R.A. (1975): Sobolev spaces. — New York: Academic Press. Bryson S. and Levy D. (2001): Central schemes for multidimensional Hamilton-Jacobi Equations. — NASA Techn. Rep., NAS-01-014. Cesari L. (1983): Optimization – Theory and Applications. — New York: Springer. Fleming W.H. and Rishel R.W. (1975): Deterministic and Stochastic Optimal Control. — New York: Springer. Jacewicz E. (2001): An algorithm for construction of -value functions for the Bolza control problem. — Int. J. Appl. Math. Comput. Sci., Vol. 11, No. 2, pp. 391–428. Karlsen K.H. and Risebro N.H. (2002): Unconditionally stable methods for Hamilton-Jacobi equations. — J. Comput. Phys., Vol. 180, No. 2, pp. 710–735.. Kurganov A. and Tadmor E. (2000): New high-resolution semidiscrete central schemes for Hamilton–Jacobi equations. — J. Comput. Phys., Vol. 160, No. 2, pp. 720–742. Szpiro A. and Dupuis P. (2002): Second order numerical methods for first order Hamilton-Jacobi equations. — SIAM J. Numer. Anal., Vol. 40, No. 3, pp. 1136–1183. Tang H.Z., Tang T. and Zhang P. (2003): An adaptive mesh redistribution method for nonlinear Hamilton–Jacobi equations in two- and three-dimensions. — J. Comput. Phys., Vol. 188, No. 2, pp. 543–572.. Received: 27 May 2004 Revised: 8 November 2004 Re-revised: 10 January 2005.

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