- Email: [email protected]
A Piecewise Linear Minimax Solution of the Hamilton-Jacobi Equation
Copyright © IFAC Nonsmooth and Discontinuous Problems of Control and Optimization, Chelyabinsk, Russia, 1998
A PIECEWISE LINEAR MINIMAX SOLUTION OF THE HAMILTON-JACOBI EQUATION
L.G. Shagalova Institute of Mathematics and Mechanics, S. K ovalevskaya str., 16, Ekaterinburg, 620219, Russia e-mail: [email protected]
Abstract: The Cauchy problem for the Hamilton - Jacobi equation with Hamiltonian independent of time and the phase variable is considered under the assumption that the Hamiltonian and the boundary function are piecewise linear. A finite algorithm for the construction of the exact piecewise linear minimax (and/or viscosity) solution is developed in the case when the phase space is twodimensional. The fact that a minimax solution is a piecewise linear function is established also for one special case when the phase space is three-dimensional. This results can be used in the researches of bifurcations of piecewise smooth solutions of PDEs of first order, and in the development of numerical methods. Copyright @1998 [FAC Keywords: algorithms; differential equations; differential games; properties; numerical methods; piecewise linear analysis.
stability
substantiated.
1. INTRODUCTION
In this paper the Cauchy problem for the Hamilton - J acobi equation with Hamiltonian independent of time and the phase variable is considered. It is also assumed that the Hamiltonian and the boundary function are piecewise linear. For the case when the phase space is two-dimensional, and for one special case when the phase space is three-dimensional it is established that under these assumptions minimax solution also turns out to be a piecewise linear function.
It is known that Hamilton-J acobi equations usually do not have classical solutions. By this reason many concepts of a generalized solution have been considered in the theory of these equations.
The investigations of many mathematicians are connected with the concept of a viscosity solution, introduced by Crandall and Lions (1983). Another approach to the definition of a generalized solution originates from the theory of positional differential games (Krasovskil and Subbotin, 1974). In (Subbotin, 1980) a generalized solution of first - order PDE was defined by substituting the equation by a pair of differential inequalities. The term "minimax solution" was suggested in (Subbotin, 1991), where a study of such solutions was given. In particular, the equivalence of minimax and viscosity solutions was proved in this book. In monograph (Subbotin, 1995) the fact, that the theory of minimax solutions can be considered as a nonclassical method of characteristics, is
If the phase space is two-dimensional, and the Hamiltonian is positively homogeneous, a finite algorithm for constructing of the exact minimax solution is elaborated and justified. The solution is constructed in the class of piecewise linear functions, that may be defined by the use of structural matrices. On the base of this algorithm the computing program and the program, drawing the level lines for corresponding minimax solutions were worked out.
The results of this paper can be used in researches
193
d+cp(y; I) = lim sup ([
of bifurcations of piecewise smooth generalized solutions to the first-order PDEs in general form and in the development of corresponding numerical methods.
dO 0<6<_
It should be noted, that the inequalities (9)(10) can be written in other equivalent forms (Subbotin, 1991,1995). Exact formulas are known
for solution of problem (7), (8) ((1), (2)) in certain cases only. For example, if the Hamiltonian H or the boundary function 0- are convex or concave, then the Hopf formulas (Hopf, 1965) can be applied directly. It follows from the results of (Pshenichnyi and Sagaidak, 1970; Bardi and Evans, 1984) that the generalized solutions defined by the Hopfformulas are viscosity (and/or minimax). However, there is no success in obtaining explicit formulas in the general case.
2. THE PROBLEM STATEMENT The following Cauchy problem is considered
8u(at t, x) +H ( Dxu (t,x ) ) =O,tE ( ) ,xERn 0,1 (1) u(l,x) = o-(x),x E Rn (2) where Dxu is the gradient of the function u with respect to the variable x, and the Hamiltonian H : Rn --+ R and the terminal function 0- : Rn --+ R are continuous and satisfy the conditions
IH(s(l») - H(s(2))1 ~
"lls(l) -
s(2)11,
(3)
o-(ax) = ao-(x) for any s(i) E Rn, X E Rn, a 2:
Below the problem (7), (8) will be considered under additional assumption that the functions H and 0- are piecewise linear.
(4) 3. THE CONSTRUCTION OF AN EXACT SOLUTION ON THE PLANE
o.
It is required to construct a minimax (and/or
In this section the problem (7),(8) is considered under the following assumptions:
viscosity) solution of this problem, which exists and is unique (Subbotin, 1991, 1995). Use the relation
H(·) E PL,o-C) E PL+,n = 2
x u(t, x) = (1 - t)u(O, -1-), t E [0,1), x E Rn (5)
Here the symbol P L denotes the set of positively homogeneous piecewise linear functions, and P L+ stands for the set of all nonnegative functions in P L. It should be noted, that the assumption of the nonnegativity on the function 0- does not lose the generality of consideration, and the positive homogeneity condition
-t
following from the positive homogeneity condition (4), then the problem (1), (2) reduces to the problem of finding the function
(6)
The function
H(as) = aH(s)
H(D -10
A finite algorithm for the construction of the exact minimax solution of the problem (7), (8) ((1), (2)) was suggested (Subbotin and Shagalova, 1992). Now this algorithm describing below is justified in full.
(8)
3.1. The elementary problems.
In essence, the algorithm consists of successive solving elementary problems arising in a definite order. These problems can be formulated as follows. Let
min [d- +H(s) -
(9) max[d+ +H(s) - 0
fERn
-
o-+(y) = max{ < a, y >, < b, y >},
(10) for any y E Rn, s E Rn. The symbols d- cp(y; I) and d+
d-cp(y; f)
= lim inf ([cp(y + 81) _10 0<0<_
(12)
for any s E R 2 , a 2: 0, is essential.
The symbol < 1, s > denotes the scalar product of the vectors 1 and s. The minimax solution of (7) is the continuous function satisfying the pair of differential inequalities fERn
(11)
o--(y) = min{< a, y >, < b,y >}, where a, band y are vectors in R 2 . Problems 1 and 2. Let some linearly independent vectors a and b be given. In problem 1 [in problem 2] it is required to construct the minimax solution
cp(y)]8- 1 ,
194
of problem (7), (8), (11) with
(7'
=
(7'+
[with
(7'
=
together of two linear functions. assertion is valid.
(7'-].
The following
The solution iP of (6), (7), (11) is formed by sewing together the linear functions
The solutions of these problems are the functions
iP+(y) = max iPl(Y), ZE[a,b]
iPl(Y) =< I, Y > +H(l), I E L,
iP-(y) = min iPl(Y),
where the set L consists of a finite number of elements, and Z CL, (L\Z) C (Z* nn).
IE[a,b]
where [a, b] = {Aa + (1- >.)bl>' E [0, I]), iPz(y) =< I, Y > +H(l). Problems 3 and 4. Let some linearly independent vectors a, b and a number r > 0 be given. Let
The solution iP is constructed in the class of functions, that are formed by "sewing together" a finite collection of simple piecewise linear functions (SPLF). The main property of an SPLF is the following. If t/J is an SPLF, then for an arbitrary point Y* in its domain, there is a neighbourhood Oe(Y*) where t/J has one of three possible representations:
iP*(y) = max{iPa(Y), iPb(Y)}, iPo(y)
=min{iPa(Y), iPb(Y)} ,
It is required to construct a continuous functions iPo and iPo satisfying the relations
{y E R 2IiPo(y) = r} = {y E R 2IiP*(Y) = r},
t/J(Y) = max{ < Si, Y > +h i , < Sj, Y > +hj >,
{y E R 2IiPO(y) < r} = {y E R 2IiP*(Y) < r} = GO, {y E R 2IiPo(y) r} {y E R 2IiP*(Y) = r}, {y E R 2IiPo(Y) < r} = {y E R 2IiP*(Y) < r} = G*,
t/J(Y) = min{ < Si, Y > +h i , < Sj, Y > +hj > . Here Si and Sj are vectors in R 2, and hi and hj are numbers. Thus the domain of definitions of an SPLF contains no points in small neighbourhoods of which three or more linear functions are sewn together.
= =
In the region GO [respectively, G*] the function iPo [iPo] is to satisfy (9) and (10).
Structural matrices may be used for formal definition of SPLFs. The structural matrix (SM) contains an information about all linear functions, that are forming the corresponding SPLF. Given the SM, one can easy calculate the value of the corresponding SPLF in every point in its domain.
The solution of Problem 3 is the function iPO(y) = max. iP.(y) for sE Sr(a, b), where Sr(a, b) = {s E con(a, b)liP.( wo) = r},
con(a, b)
= {Aa + Jlbl>' 2:: 0, Jl 2:: O}
In the region {y E R 2IiP(Y) > O} the solution iP of (6), (7), (11) can be written with the help of the sequence of structural matrices
and the point Wo is the solution of the system of two linear equations
00),M2(C2, cI), ... , Mk(Ck, Ck-l), (13) Ck < Ck-l < ... < Cl < Co = 00
Ml(Cl,
o~ In the general case (i.e., for arbitrary a, b, and r) the solution of Problem 4 may fail to exist. However, in the cases which arise in the construction of the solution of (7), (8), (11) the function iPo exists and has the form
The matrix Mi(Ci, ci-I) defines the function iP in the region, where Ci < iP < Ci-l· In the domain of iP there are finite number of singular points at which at least three linear functions are sewing together. All of such points are situated on the level lines {y E R 2IiP(Y) = cd, i = 1, ... , k, and for every i E {I, ... , k} there is at least one singular point on the line {y E R 2IiP(Y) = cd. To calculate any number Ci, i = 1, ... , k, called critical level, it is necessary to solve a finite number of Cramer systems of three linear equations.
3.2. The structure of the solution.
The function (7', according to (11), is formed by "sewing together" a finite collection of linear functions (7'i(Y) =< Si,Y >,i = 1, ... ,n e . Let Z = {sdi = 1, ..., ne}, Z* = co(Z U{O}), where o and coN denote the zero vector and the convex hull of the set N. Let also the symbol n denote the set of points where the piecewise linear function H is not differentiable. Thus the set n consists of the point 0 and the points at which H is the sewing
Thus, formally the algorithm consists of a finite number of steps. Constructing of some SPLF is a matter of every step. The number of steps (the number of matrices in the sequence (13) is not known in advance. It will be defined in the process of the construction.
195
H (s)
= min < s, p > + max < s, q >, s E R 2 , pEP qEQ
where P and Q are convex polygons. Some illustrations, obtained with the help of these programs, are represented on Fig. l.
(a)
On the left side of every picture on Fig. 1 there is the line {x E R 2 1u(x) = 1}. The origin is marked by cross. On the right side of the same picture there are level lines of corresponding minimax solution. The dotted lines denote the critical levels. Also it should be noted that the coordinate systems (the origins, and the scales on the axes) for the left and right parts of the picture are different, because the drawing program chooses them automatically. 3.4. The solution of non-reduced problem.
Constructing the solution i{! of reduced problem (7), (8), (11), one can restore the solution u of original problem (1), (2), (11). This solution is a piecewise linear function defined in the space of variables t and x.
(b)
The following fact is valid. Let the sequence of structural matrices (13) define the solution problem (7), (8), (11). Then for any t" E (0,1) the function u(t·, x) can be written in the form
h \!
Mi( ci, (0), Mi(c;, ci), ... , Mk(c k, ck_l)' ~ ck < ck- 1 < ... < ci < Co = 00,
°
where cT = cd(l - t.), and the matrix Mt can be written formally with the help of simple transformation of the matrix M; according with a definite rules.
(c)
4. THE STRUCTuRE OF THE SOLUTION IN SOME SPECIAL CASES Let the Hamiltonian H in problem (1), (2) be not positively homogeneous, and let u(t, x) be the solution of this problem. Define the positively homogeneous function _ _ _ _ _ _ _ 11
if
-----------;111: :
r
#-
° (14)
if
r
= 0,
where s E Rn, r E R.
Fig. 1. Level lines of terminal functions and corresponding minimax solutions.
Consider the following Caushy problem
au#
7ft + H
3.3. The computer realization.
On the base of the above algorithm the computing program and the program, drawing the level lines for corresponding minimax solutions were worked out for the case, when the Hamiltonian is the function of the form
#
# au#
(Dxu ,
ay )-_ 0,
u(l, x, y) = u(x)
+ y.
(15)
(16)
Here t E [0, l],x E Rn,y ER. The following assertion is valid (Subbotin, 1991).
196
Pshenichnyi, B.N. and M.I. Sagaidak (1970). Differential games of prescribed duration. Kibernetika, 2, pp. 54-63. (in Russian; English trans!., (1970). Cybernetics, 6, pp. 72-
The function u(t,x) is the solution of (1), (2) iff the function u#(t, x, y) = u(t, x)+y is the solution of (15), (16). Let the functions Hand u be piecewise linear, and, besides, u satisfy the condition (4). Using the above assertion, one can prove that minimax solution of (1), (2) is a piecewise linear function also in the cases (a) n = 2, and H is not positively homogeneous; (b) n = 3, H is positively homogeneous, and u is a function of the form u( x) = u( Xl, X2) + Xs, where X = (Xl, X2, xs)T E R S , and the symbol T denotes transposition.
83.)
Subbotin, A.I. (1980). A generalization of the basic equation of the theory of differential games. Dokl. Akad. Nauk SSSR. 254, pp. 293-297. (in Russian; English trans!., (1980). Soviet Math. DoH., 22, pp. 358-362.) Subbotin, A.I. (1991). Minimax inequalities and Hamilton-Jacobi equations. Nauka, Moscow. Subbotin, A.I. (1995). Generalized solutions of first order PDEs: the dynamical optimization perspective. Birkhiiuser, Boston. Subbotin, A.I. and L.G. Shagalova (1992). A piecewise linear solution of the Cauchy problem for the Hamilton-jacobi equation. Ross. Akad. Nauk Doklady, 325 (5), pp. 932-936. (in Russian; English transl., (1993). Russian Acad. Sci. Dokl. Math., 46 (1), pp.
The proof is based on solving of elementary problems analogous those considered in section 3.1.
5. CONCLUSION The Caushy problem for the Hamilton - J acobi equation with Hamiltonian independent of time and the phase variable was considered in this article under the assumption that the data are piecewise linear. A finite algorithm for the construction of the exact minimax solution on the plane was described. Special case when minimax solution is piecewise linear was picked out when the phase space is three-dimensional. This results can be used in the researches of bifurcations of piecewise smooth solutions of first order PDEs in general form, and for approximations of such solutions.
144-148.)
ACKNOWLEDGEMENTS This research was supported by the Russian Foundation of Basic Researches under Grant No. 96-01-00219 and Grant No. 96-15-96245.
REFERENCES Bardi, M. and L.C. Evans (1984). On Hopf's formulas for solutions of Hamilton-Jacobi equations. N onlinear Analysis, Theory, Methods, App/.,8 (11), pp. 1373-1381. Crandall, M.G. and P.L. Lions (1983). Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc., 277 (1), pp. 1-42. Hopf, E. (1965). Generalized solutions of nonlinear equations of first order. J. M ath. Mech. 14, pp. 951-973. Krasovskil, N.N. and A.I. Subbotin (1974). Positional differential games. Nauka, Moscow. (in Russian; rev. English trans!., (1988). Game-theoretic control problems. Springer-Verlag, Berlin. 197
A PIECEWISE LINEAR MINIMAX SOLUTION OF THE HAMILTON-JACOBI EQUATION
L.G. Shagalova Institute of Mathematics and Mechanics, S. K ovalevskaya str., 16, Ekaterinburg, 620219, Russia e-mail: [email protected]
Abstract: The Cauchy problem for the Hamilton - Jacobi equation with Hamiltonian independent of time and the phase variable is considered under the assumption that the Hamiltonian and the boundary function are piecewise linear. A finite algorithm for the construction of the exact piecewise linear minimax (and/or viscosity) solution is developed in the case when the phase space is twodimensional. The fact that a minimax solution is a piecewise linear function is established also for one special case when the phase space is three-dimensional. This results can be used in the researches of bifurcations of piecewise smooth solutions of PDEs of first order, and in the development of numerical methods. Copyright @1998 [FAC Keywords: algorithms; differential equations; differential games; properties; numerical methods; piecewise linear analysis.
stability
substantiated.
1. INTRODUCTION
In this paper the Cauchy problem for the Hamilton - J acobi equation with Hamiltonian independent of time and the phase variable is considered. It is also assumed that the Hamiltonian and the boundary function are piecewise linear. For the case when the phase space is two-dimensional, and for one special case when the phase space is three-dimensional it is established that under these assumptions minimax solution also turns out to be a piecewise linear function.
It is known that Hamilton-J acobi equations usually do not have classical solutions. By this reason many concepts of a generalized solution have been considered in the theory of these equations.
The investigations of many mathematicians are connected with the concept of a viscosity solution, introduced by Crandall and Lions (1983). Another approach to the definition of a generalized solution originates from the theory of positional differential games (Krasovskil and Subbotin, 1974). In (Subbotin, 1980) a generalized solution of first - order PDE was defined by substituting the equation by a pair of differential inequalities. The term "minimax solution" was suggested in (Subbotin, 1991), where a study of such solutions was given. In particular, the equivalence of minimax and viscosity solutions was proved in this book. In monograph (Subbotin, 1995) the fact, that the theory of minimax solutions can be considered as a nonclassical method of characteristics, is
If the phase space is two-dimensional, and the Hamiltonian is positively homogeneous, a finite algorithm for constructing of the exact minimax solution is elaborated and justified. The solution is constructed in the class of piecewise linear functions, that may be defined by the use of structural matrices. On the base of this algorithm the computing program and the program, drawing the level lines for corresponding minimax solutions were worked out.
The results of this paper can be used in researches
193
d+cp(y; I) = lim sup ([
of bifurcations of piecewise smooth generalized solutions to the first-order PDEs in general form and in the development of corresponding numerical methods.
dO 0<6<_
It should be noted, that the inequalities (9)(10) can be written in other equivalent forms (Subbotin, 1991,1995). Exact formulas are known
for solution of problem (7), (8) ((1), (2)) in certain cases only. For example, if the Hamiltonian H or the boundary function 0- are convex or concave, then the Hopf formulas (Hopf, 1965) can be applied directly. It follows from the results of (Pshenichnyi and Sagaidak, 1970; Bardi and Evans, 1984) that the generalized solutions defined by the Hopfformulas are viscosity (and/or minimax). However, there is no success in obtaining explicit formulas in the general case.
2. THE PROBLEM STATEMENT The following Cauchy problem is considered
8u(at t, x) +H ( Dxu (t,x ) ) =O,tE ( ) ,xERn 0,1 (1) u(l,x) = o-(x),x E Rn (2) where Dxu is the gradient of the function u with respect to the variable x, and the Hamiltonian H : Rn --+ R and the terminal function 0- : Rn --+ R are continuous and satisfy the conditions
IH(s(l») - H(s(2))1 ~
"lls(l) -
s(2)11,
(3)
o-(ax) = ao-(x) for any s(i) E Rn, X E Rn, a 2:
Below the problem (7), (8) will be considered under additional assumption that the functions H and 0- are piecewise linear.
(4) 3. THE CONSTRUCTION OF AN EXACT SOLUTION ON THE PLANE
o.
It is required to construct a minimax (and/or
In this section the problem (7),(8) is considered under the following assumptions:
viscosity) solution of this problem, which exists and is unique (Subbotin, 1991, 1995). Use the relation
H(·) E PL,o-C) E PL+,n = 2
x u(t, x) = (1 - t)u(O, -1-), t E [0,1), x E Rn (5)
Here the symbol P L denotes the set of positively homogeneous piecewise linear functions, and P L+ stands for the set of all nonnegative functions in P L. It should be noted, that the assumption of the nonnegativity on the function 0- does not lose the generality of consideration, and the positive homogeneity condition
-t
following from the positive homogeneity condition (4), then the problem (1), (2) reduces to the problem of finding the function
(6)
The function
H(as) = aH(s)
H(D
A finite algorithm for the construction of the exact minimax solution of the problem (7), (8) ((1), (2)) was suggested (Subbotin and Shagalova, 1992). Now this algorithm describing below is justified in full.
(8)
3.1. The elementary problems.
In essence, the algorithm consists of successive solving elementary problems arising in a definite order. These problems can be formulated as follows. Let
min [d-
(9) max[d+
fERn
-
o-+(y) = max{ < a, y >, < b, y >},
(10) for any y E Rn, s E Rn. The symbols d- cp(y; I) and d+
d-cp(y; f)
= lim inf ([cp(y + 81) _10 0<0<_
(12)
for any s E R 2 , a 2: 0, is essential.
The symbol < 1, s > denotes the scalar product of the vectors 1 and s. The minimax solution of (7) is the continuous function satisfying the pair of differential inequalities fERn
(11)
o--(y) = min{< a, y >, < b,y >}, where a, band y are vectors in R 2 . Problems 1 and 2. Let some linearly independent vectors a and b be given. In problem 1 [in problem 2] it is required to construct the minimax solution
cp(y)]8- 1 ,
194
of problem (7), (8), (11) with
(7'
=
(7'+
[with
(7'
=
together of two linear functions. assertion is valid.
(7'-].
The following
The solution iP of (6), (7), (11) is formed by sewing together the linear functions
The solutions of these problems are the functions
iP+(y) = max iPl(Y), ZE[a,b]
iPl(Y) =< I, Y > +H(l), I E L,
iP-(y) = min iPl(Y),
where the set L consists of a finite number of elements, and Z CL, (L\Z) C (Z* nn).
IE[a,b]
where [a, b] = {Aa + (1- >.)bl>' E [0, I]), iPz(y) =< I, Y > +H(l). Problems 3 and 4. Let some linearly independent vectors a, b and a number r > 0 be given. Let
The solution iP is constructed in the class of functions, that are formed by "sewing together" a finite collection of simple piecewise linear functions (SPLF). The main property of an SPLF is the following. If t/J is an SPLF, then for an arbitrary point Y* in its domain, there is a neighbourhood Oe(Y*) where t/J has one of three possible representations:
iP*(y) = max{iPa(Y), iPb(Y)}, iPo(y)
=min{iPa(Y), iPb(Y)} ,
It is required to construct a continuous functions iPo and iPo satisfying the relations
{y E R 2IiPo(y) = r} = {y E R 2IiP*(Y) = r},
t/J(Y) = max{ < Si, Y > +h i , < Sj, Y > +hj >,
{y E R 2IiPO(y) < r} = {y E R 2IiP*(Y) < r} = GO, {y E R 2IiPo(y) r} {y E R 2IiP*(Y) = r}, {y E R 2IiPo(Y) < r} = {y E R 2IiP*(Y) < r} = G*,
t/J(Y) = min{ < Si, Y > +h i , < Sj, Y > +hj > . Here Si and Sj are vectors in R 2, and hi and hj are numbers. Thus the domain of definitions of an SPLF contains no points in small neighbourhoods of which three or more linear functions are sewn together.
= =
In the region GO [respectively, G*] the function iPo [iPo] is to satisfy (9) and (10).
Structural matrices may be used for formal definition of SPLFs. The structural matrix (SM) contains an information about all linear functions, that are forming the corresponding SPLF. Given the SM, one can easy calculate the value of the corresponding SPLF in every point in its domain.
The solution of Problem 3 is the function iPO(y) = max. iP.(y) for sE Sr(a, b), where Sr(a, b) = {s E con(a, b)liP.( wo) = r},
con(a, b)
= {Aa + Jlbl>' 2:: 0, Jl 2:: O}
In the region {y E R 2IiP(Y) > O} the solution iP of (6), (7), (11) can be written with the help of the sequence of structural matrices
and the point Wo is the solution of the system of two linear equations
00),M2(C2, cI), ... , Mk(Ck, Ck-l), (13) Ck < Ck-l < ... < Cl < Co = 00
Ml(Cl,
o~ In the general case (i.e., for arbitrary a, b, and r) the solution of Problem 4 may fail to exist. However, in the cases which arise in the construction of the solution of (7), (8), (11) the function iPo exists and has the form
The matrix Mi(Ci, ci-I) defines the function iP in the region, where Ci < iP < Ci-l· In the domain of iP there are finite number of singular points at which at least three linear functions are sewing together. All of such points are situated on the level lines {y E R 2IiP(Y) = cd, i = 1, ... , k, and for every i E {I, ... , k} there is at least one singular point on the line {y E R 2IiP(Y) = cd. To calculate any number Ci, i = 1, ... , k, called critical level, it is necessary to solve a finite number of Cramer systems of three linear equations.
3.2. The structure of the solution.
The function (7', according to (11), is formed by "sewing together" a finite collection of linear functions (7'i(Y) =< Si,Y >,i = 1, ... ,n e . Let Z = {sdi = 1, ..., ne}, Z* = co(Z U{O}), where o and coN denote the zero vector and the convex hull of the set N. Let also the symbol n denote the set of points where the piecewise linear function H is not differentiable. Thus the set n consists of the point 0 and the points at which H is the sewing
Thus, formally the algorithm consists of a finite number of steps. Constructing of some SPLF is a matter of every step. The number of steps (the number of matrices in the sequence (13) is not known in advance. It will be defined in the process of the construction.
195
H (s)
= min < s, p > + max < s, q >, s E R 2 , pEP qEQ
where P and Q are convex polygons. Some illustrations, obtained with the help of these programs, are represented on Fig. l.
(a)
On the left side of every picture on Fig. 1 there is the line {x E R 2 1u(x) = 1}. The origin is marked by cross. On the right side of the same picture there are level lines of corresponding minimax solution. The dotted lines denote the critical levels. Also it should be noted that the coordinate systems (the origins, and the scales on the axes) for the left and right parts of the picture are different, because the drawing program chooses them automatically. 3.4. The solution of non-reduced problem.
Constructing the solution i{! of reduced problem (7), (8), (11), one can restore the solution u of original problem (1), (2), (11). This solution is a piecewise linear function defined in the space of variables t and x.
(b)
The following fact is valid. Let the sequence of structural matrices (13) define the solution problem (7), (8), (11). Then for any t" E (0,1) the function u(t·, x) can be written in the form
h \!
Mi( ci, (0), Mi(c;, ci), ... , Mk(c k, ck_l)' ~ ck < ck- 1 < ... < ci < Co = 00,
°
where cT = cd(l - t.), and the matrix Mt can be written formally with the help of simple transformation of the matrix M; according with a definite rules.
(c)
4. THE STRUCTuRE OF THE SOLUTION IN SOME SPECIAL CASES Let the Hamiltonian H in problem (1), (2) be not positively homogeneous, and let u(t, x) be the solution of this problem. Define the positively homogeneous function _ _ _ _ _ _ _ 11
if
-----------;111: :
r
#-
° (14)
if
r
= 0,
where s E Rn, r E R.
Fig. 1. Level lines of terminal functions and corresponding minimax solutions.
Consider the following Caushy problem
au#
7ft + H
3.3. The computer realization.
On the base of the above algorithm the computing program and the program, drawing the level lines for corresponding minimax solutions were worked out for the case, when the Hamiltonian is the function of the form
#
# au#
(Dxu ,
ay )-_ 0,
u(l, x, y) = u(x)
+ y.
(15)
(16)
Here t E [0, l],x E Rn,y ER. The following assertion is valid (Subbotin, 1991).
196
Pshenichnyi, B.N. and M.I. Sagaidak (1970). Differential games of prescribed duration. Kibernetika, 2, pp. 54-63. (in Russian; English trans!., (1970). Cybernetics, 6, pp. 72-
The function u(t,x) is the solution of (1), (2) iff the function u#(t, x, y) = u(t, x)+y is the solution of (15), (16). Let the functions Hand u be piecewise linear, and, besides, u satisfy the condition (4). Using the above assertion, one can prove that minimax solution of (1), (2) is a piecewise linear function also in the cases (a) n = 2, and H is not positively homogeneous; (b) n = 3, H is positively homogeneous, and u is a function of the form u( x) = u( Xl, X2) + Xs, where X = (Xl, X2, xs)T E R S , and the symbol T denotes transposition.
83.)
Subbotin, A.I. (1980). A generalization of the basic equation of the theory of differential games. Dokl. Akad. Nauk SSSR. 254, pp. 293-297. (in Russian; English trans!., (1980). Soviet Math. DoH., 22, pp. 358-362.) Subbotin, A.I. (1991). Minimax inequalities and Hamilton-Jacobi equations. Nauka, Moscow. Subbotin, A.I. (1995). Generalized solutions of first order PDEs: the dynamical optimization perspective. Birkhiiuser, Boston. Subbotin, A.I. and L.G. Shagalova (1992). A piecewise linear solution of the Cauchy problem for the Hamilton-jacobi equation. Ross. Akad. Nauk Doklady, 325 (5), pp. 932-936. (in Russian; English transl., (1993). Russian Acad. Sci. Dokl. Math., 46 (1), pp.
The proof is based on solving of elementary problems analogous those considered in section 3.1.
5. CONCLUSION The Caushy problem for the Hamilton - J acobi equation with Hamiltonian independent of time and the phase variable was considered in this article under the assumption that the data are piecewise linear. A finite algorithm for the construction of the exact minimax solution on the plane was described. Special case when minimax solution is piecewise linear was picked out when the phase space is three-dimensional. This results can be used in the researches of bifurcations of piecewise smooth solutions of first order PDEs in general form, and for approximations of such solutions.
144-148.)
ACKNOWLEDGEMENTS This research was supported by the Russian Foundation of Basic Researches under Grant No. 96-01-00219 and Grant No. 96-15-96245.
REFERENCES Bardi, M. and L.C. Evans (1984). On Hopf's formulas for solutions of Hamilton-Jacobi equations. N onlinear Analysis, Theory, Methods, App/.,8 (11), pp. 1373-1381. Crandall, M.G. and P.L. Lions (1983). Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc., 277 (1), pp. 1-42. Hopf, E. (1965). Generalized solutions of nonlinear equations of first order. J. M ath. Mech. 14, pp. 951-973. Krasovskil, N.N. and A.I. Subbotin (1974). Positional differential games. Nauka, Moscow. (in Russian; rev. English trans!., (1988). Game-theoretic control problems. Springer-Verlag, Berlin. 197