VieinamJ Math. (2015)43 459-469 DOI 10.I007/SIOOI3-OI5-0132-4
Construction of Restricted Controls for a Non-equilibrium Point in Global Sense
Valery I. Korobov • Vasyl O. Skoryk
Received; !5 November 2013 / Accepled: 2 December 2014 / Published online 27 February 2015
© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Smgapore 2015
Abstract The paper is devoted to the problem of construction of a constrained control, which transfers a control system from any point to a given non-equilibnum point in a finite time in global sense. The construction of the mentioned control is based on the control- lability function method. The problem is solved for linear systems. The obtained result is illustrated with model example.
Keywords Linear control system • Controllability function method • Non-equilibrium point
Mathematics Subject Classification (2000) 34H05 • 93B05 93C05 • 93C10
t Introduction For the system
-v = fix, ll). X e R". u eQ d M^ (1) with constraints on the conlrol ofthe form H e P — i« G R*" : ||»!l ^ d], we consider
(he problem of construction of a control uit). which transfers any point JCQ e K" to a given non-equiiibrium point .v, e R" ofthe system (1) in a fi^e finite time T = Tixo.x^) and
Dedicated lo Professor Nguyen Khoa Son V. I. Korobov (»:) • V. O Skoryk
V. N Karazin Kharkov National University, sq. Svobody, 4, 61022, Kharkov, Ukraine e-mail vkorobov@univerkharkovua
VO Skoryk
e-mail skoryk@univerkharkovua V. 1. Korobov
Insuiule of Mathemaiics, S/c/.ecin University, Wieikopolska str 15. Szczecin, 70-451. Poland e-mail: [email protected],pl
^ Spru
V I. Korobov. V O. Skoryk satisfies given constraints ||M(r) || < d for all r e [0, T], where t/ > 0 is a given number. Let the system (1) be a linear system of the form
i = Ax + Bu, xeM", H e ^ CM", (2) for which the condition iank(S, AB, . , A " ~ ' J 5 ) — n holds. Let u(r) be a control, which
transfers the point XQ to the point x , in a finite time T along the trajectory xU) of system (2). Then,
xit) - e^' (xa -F j e-^^Buiz)dz\ , and using xiT) — J:*, we obtain the equality
U^xo-e-^' Xi,+ e-^^Buiz)dt. (3) Jo
Equality (3) means that the control uU) transfers the point zo — xo — e~^^X:f to the ongin in the time T However, since T is an object to be found, then the point zo is unknown.
Thus, we need to find the time T and construct the control u(t), which transfers so to the ongin in time T and satisfies the preassigned constraints.
We will investigate the problem using the controllability function method, which was proposed by Korobov m [1]. Following the method, for system (1), we introduce the controllability function 0 ( J : ) ( 0 ( J ; ) > 0 at X / 0 and 0 ( 0 ) - 0) and the control uix) = uix, &ix)) so that the differential inequality
A a&ix)
Yl -^^f-(^' "(^)) 5 -Vi&M) W
^ ox,
holds [2], where epi©) > 0 at 0 ?i 0, ^(0) = 0, and / Q " ^ < CO (O > 0). Inequality (4) means that the control H(X) is chosen so that the trajectory follows the direction of decrease of the function 0 ( x ) . Due to the properties of the function cpi&), this inequality ensures that the trajectory hits the ongin in a finite time. In [1-3], tpi0) ~ ^ 0 ' ~ " , hence (4) takes the form
• A 90(x) , I
E -^f'i^- «U)) < -^&'-^ix), ^ > 0, « > 0, (5)
,=1 "^^^
Then the time ofthe motion satisfies the estimate T{x(j) < | 0 a (xo). In particular, a case when one can give the precise value of die time of motion is of great interest. This is possible 'f E',L| ^ / ^ U . uix)) = -1 Then r(:.o) - 0(x(,).
In the controllability function method, the controllability function is defined implicitly as the positive solution ofthe equation 0 ( 0 , x) — 0 For example, for the canonical linear system, the function 0 i&, x) is a polynomial with respect to & and x. This is one of the special features in which the controllability function method and the Lyapunov function method are different, since traditionally the explicit form of representation of the Lyapunov function IS used Note, however, that for a linear system in the case a- — oo the equation 0i0.x) ^ 0 turns into the expiicit equation &ix) ^ V(jc), where V(x) is the Lyapunov function for the closed system.
The paper is organized as follows. In Section 2, the problem of construction of a control, which transfers any given point to any given non-equilibnum point m a finite time and satisfies the preassigned constraints, for a linear system is considered. Namely, the problem in global sense for the linear system (2) with the matrix A, which has eigenvalues with non- positive real parts, is solved. The construction of such control is based on the controllability
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ConsUiiction of Restncted Controls for a Non-equtlibrium Point
function method [1, 2]. The main result is formulated in Theorem 1. The obtained results are illustrated wilh a model example. In Section 3 (Conclusion), we indicate a direction of the future development of the obtained results in the given paper.
2 Solution of the Problem for Linear Systems in Global Sense
Consider the problem of construction ofa control w(/), which transfers any point Xo e R"
to any given non-equilibrium point x, e R" of the system (2) in a fi-ee finite time T — Tixo. X*) and satisfies the preassigned constraints [|M(?)|[ < d for all t e [0, T], Using the controllability function method, we give the solution of the problem for the system (2) in global sense in the following theorem.
Theorem 1 Assume that in the system (2), the matrix A has eigenvalues wilh non-positive real parts, and the condition rank(fi. AB A"~^ B) = n holds. Assume thai for a given poinlxo € R" and for a given point .X., e R", which is a non-equilibrium point of the system (2), the lime T is a positive soluUon ofthe equation
2aoT - [N-\T) (XO - e-^'^x^ . xo - e'^^x^) - 0, (6) where
0 < « o < 2 r f - m i n | / , j / / | | 5 | | M , / > 0. y > 0. (7) N-'iT) is the inverse matrix to NiT) = fjil - ^)e-'^'BB*e-'^''dt, T > 0.
Then the control uU) of ihe form
uit) = --B*N-\T - t)zit), t e [0, T], (8) transfers the point xo to the point x, in the time T along the trajectory
x ( r ) = z a ) + e - ' * ' ^ - ' * x * , (9) where zU) is the solution ofthe Cauchy problem
{ i ^ U - ^BB*N-UT -t)\z.
[ziO)=xo-e-^'''x^.
and satisfies the preassigned constraints \\uil) || < dfor all I e [0, T].
(10)
Proof The proof is conducted on the basis of results of the paper [3]. Fnstly, according to the controllability function method, we show t h a t ^ j ' any positive number ao the equation
* { 0 , , - ) = O , z e a " \ { 0 ) . (11) where 0 ( 0 , z) = 2ao& — (/V~' ( 0 ) z , z) has a unique positive solution &iz), which is
continuously differentiable function for alt z ^ Oand&iz) is a continuous function at z — 0 if 0iO) = 0.
Since g ^ A ' ( 0 ) = Ni&), where Ni@) ^ ^_ fo le'^'BB*e-'^''dl is a positive definite matnx for 0 > 0, then
— CPU'), z) = 2ao + (N-\&)'Ni0)N-'i0)z.z\ > 2«o > 0, 0 > 0. (12)
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462 V. I. Korobov. V. 0 Skoryk Since IliVOJII < IIBII^ ( « ' " " » - 1 ) / ( 2 | | A | | ) , then
*((=),,-)< 2xne - J i L s-2a,e- ..^i'll'i'il'iM" , , for all z e R" \ {0).
Hence,
lim 0 ( 0 , 7 ) - - c o foraU z e R " \ | 0 | , (13)
&->^o
Since - (W~' ( 0 ) z , z) is an increasing function with respect to &, and
- ( - ' < « > " ) - ^ - S ^ ^ P l ^ ..an„M.M„,,
then the function - (A'"' i&)z, z) has a finite limit as 0 ^ -|-oo for all z e M" \ (0). Hence
hm 0 ( 0 , z) - +00 for all z e M" \ (O). (14) C-J^+oa
From (12)-(!4), it follows that (11) has a unique positive solution 0 ( z ) for all z e W \ (0} Since 0 ( 0 , z) is a continuously differentiable function with respect to 0, z, and g ^ 0 ( 0 , z) ?^ 0 for all z e M" \ (0), then using the implicit function theorem, we obtain that 0 ( z ) is a continuously differentiable function for all z e M" \ (Oj.
We show that the controllability function 0 ( s ) is a continuous function at z = 0 if 0 ( 0 ) ^ 0. Assume the opposite, let 0 ( z ) > e > 0 for all z e t/(0, 5) ^ {z e ffi" : llsll < b]. Since (A'~^(0)z, z) is a decreasing function with respect to 0 , then we have the inequality ( A ' - ' ( 0 ( z ) ) z , z) < (/V"-'(£)z.z) for 0 ( z ) > s. Choose 5 ^ v'2«oe/l|A'"-'(£)l|.
Then
2aQe <2aQ&(z) < ( A ' - ' ( f ) z , z ) < ||A'-'(e)ll llsll^ < 2^0^, zeUiO,^), and we obtain a contradiction Thus, 0 ( z ) < £ for all z e UiO, S).
Using the controllability function, we construct the control H'(Z) of the form
uiz)^-B~N-'i0iz))z. z e R " \ { 0 ) . (15) The control M(Z) is a Lipschitz funcuon in each set r ( / 3 | , p 2 ) = [z eR" : 0 <: p] •< \\z\\ <
pl] with a constant Z.u(e,p2) (0 < e < p2) such that £.„(£,/J2) -> + 0 0 as e ^ -1-0.
We show that the derivative of the controllability function with respect to the system Z — Az -I- Buiz) has theform
0iz) = -l for all z e K " \ ( 0 } . (16) From equality (II) at 0 — 0 ( z ) , we obtain
0(z) f ^ ( w - ' (0(z))z, z) + (W(0(z)) cpiz), cpiz))\
(iN-\0{z))A + A^N-^ i0iz))z, E)) - (BB'cpiz), cpiz)), (17) where cpiz) ^ ,/V"^'(0(s))z- Since AiV(0) -|- /V(0)A ^ B S ' - / f ( 0 ) , where A'(0) =
^ /o ^~'^' BB*e~^*'dt is a positive definite matnx, then
^ - ' ( 0 ) -I- A " A ' - ' ( 0 ) - i V - ' ( 0 ) {BB* - N(0)) A ' - ' ( 0 ) . (\%)
fi Springer
Construction of Restricted Controls for a Non-equilibnum Point From equahty (17), using equality (18), we obtain
. . . iNi0iz))cpiz).cpiz)) ( ( © b ^ ^ ^ f ^ W + A'(0(z))) cpiz), ipiz))
z e M" \ [0}
Thus, since ^ A'(0) -^ A'(0) ^ ^ ( 0 ) , dien equahty (16) follows from equality (19), i. e., in inequality (5) the equality a t « — ^ — 1 holds.
From (16), we obtain that T = 0(zo) is the time of the motion from any initial pomt zo to the origin, where 0(zo) is the unique positive solmion of (11) atz — zo-
Further, we show that m (11) a number ao can be chosen in the way that the control uiz) fromilS) satisfies the preassigned constraints ||i((z)|| < dfor all z e M" \ {0}.
Let m be the degree of the mimmal polynomial of the matrix A. Then, using Lemma 1 ofthe paper [3], there exists a constant c > 0 such that the inequahty
- iNi0)cpiz). cpiz)) > I Y^ 02*||B*A*Va)ll^ for 0 < 0 < c, (20) where / > 0, holds.
Consider the domain Q — (j e R" : 0 ( z ) < c). Using (11), (20), from (15), we obtain
\\B*<piz)fao0iz)
\[uiz)f - T l i e X z ) ! ! "
^i^ii(c-yi.Z))ipi,Z), ipyzfj
tor all z e e \ (0).
2iMieiz))viz),<piz)) BVWIPoo
2'Ei5i'e^'WIIB*'i'VMII^ 2;
Hence, we have
lSiz)\t^ii for all z e e \ (01 if 0 < ao < 2W^ (21) Now, we show that the control uiz) satisfies the constraints \\uiz)\\ < (/ for ah z € Q, = (z e E" : 0iz) >c]. Using the form of the matrix Ni&), wehave
iNi0),piz), lei!)) i i(F(e/2)9>(z), y(z)), 0 > 0, (22) where Fi0) — /Q" e^^' BB*e^'^ 'dt is a positive definite matrix by virtue of the condition rank(6, AB, .. A"-'B) ^ n. Put 0o ^ c/(2m + 2) Then
100 < 0 / 2 <is + 1)00 for .1 > m + 1, (23) where i is a natural number. Since
iFi0/2)f>iz),,piz)) > (F(seo)?)(z),y(z))
= '•£ {Fie«)e-'-0'^v(z). .-'"^'Vcz))
k^O
>/™„E||.-'''*Vz)f
where f^,a is the mimmai eigenvalue of the matnx f (0o), then using (23) from (22), we obtain the inequality
^ ( « ( 0 ( z ) ) . ( z ) , ^ W ) > { = ; ^ i | | | . - ^ - « » V z ) | f Z E C , (24)
0 Spnnger
464 V. I Korobov, V O. Skoryk Let - A J be the Jordan fonn of the matrix - A * , and S be the matnx, which reduces the matrix - A * to the matnx -A). Hence, e"''^*^« = S-'f-''"*®"5. Then, from (24), we obtain the inequality
^(«(0(z),.(z)..(z)). ^"l^ViLriglk'"'"'^'"!' <^«
for a l l s e Qi, where Jmin is the minimal eigenvalue of the matrix 5*5, ^ ( z ) — Scpiz).
Let pbe the number of Jordan boxes of the matrix —A^j, andn/ be the dimension of the /th Jordan box, / — 1 p. Using the non-negativity of real parts of eigenvalues of the matnx —A*, we obtain the inequality
^E|h-=«Hf.^;^E|E YlfjlOokV (26)
where f is the component of the vector f conesponding the (y-l-l)th position of the /th Jordan box, / — 1,. ., p. j = 0... R / _ I .
Put f'—a'+ ip' Then, using Lemma 2 ofthe paper [2] from (26), we obtain
Y.'-'j'-'^M'] +\Y.f'j(e>ak)-
^ ' 1=1 j=0 ^ (-1)
q,n 1 C/mS-„ , ( m - 1 ) ! ( m - 1 ) ! where .v^jj' is the minimal eigenvalue of the matrix ( S " ' ) * 5 - obtain
^TTl,Nfi0iz))ipiz),<piz)) i ylipiz)l-.
where y ^ /ni.n^min9m5^7n"/(m!4(m + 2)0o) > 0.
Now, we show that ||«(z) II < ( / f o r a l U e Gi. From (15), u
||Tj(-,j|2 , ll«ll^llf(z)ll^<lo0(z) , llSlpao
""*^'" - 2iNi0iz))<piz),<piz)) - 2y
(27) . Using (27) from (25). we Z E G I , (28)
ing (11), (28), we obtain z e Q, Then
IP(z)IISrf f o r a U z s Q i if 0 < oo < 2 y r f V l | B | | l (29) Choosing a number no from the condmon (7), from (21), (29), we obtain
ll«(z)ll < ll forail z s K" \ {0|. (30) Thus, using the controllability function method, we have shown that if in (11) the number
no satisfies the condition (7), then the control S-(z) ofthe form (15) transfers any initial point zo s S " \ (0) to the origin in the finite hme T = 0 ( z „ ) . where 0 ( z „ ) is the unique positive root of (11) atz - zo. and satisfies the preassigned constraints (30).
Finally, we show that Ihe conlrol (8) Iramfers Ihe point x,, a S.'to the point x eW in some finite lime T along llie Irojeaoiy xit) of Ihe fonn (9), nhere zU) is the solution ofthe Cauchy problem (10) and satisfies the preassigned constraints \\uil)\\ < d for all t e \Q Ti a Spnnger
Constniciion of Restncted Controls for a Non-equilibrium Point
Put Zo = Xo - e-'^^x^. Then (6) follows from (11) at 0 ^ T. i = zo- We choose a number ao from the condition (7) and find a positive root T of (6). Existence of a positive root of (6) follows from an existence ofthe positive root of (11). From equality (16), we obtam 0(z(f)) = - 1 . Hence, 0 ( z ( O ) = 7 " - r . Then, fi^m (15), we obtain die control«(/) of the form (8). The control i((f) transfers the point zo to the origin in the time T along the trajectory r(f), and from (30) it follows that ||H(f)|| < (/for all f e [0. T].
We show that the control uU) ofthe form (8) transfers the point AQ to the point .r* along trajectory j:(f) ofthe form (9). In fact, since
zU)=e^' lxo-e''^'^x^+ j e-"^'Buiz)dT\ = xit)
then A(f) = zU) + e-'^'^^'^-'h^. Since z(0) = zo. ^(7") - 0 then ;((0) ^ xo, xiT) ^ x,.
Theorem I is proved. D Corollary 1 Lei XQ e M" be an arbitrary point and lei ;c* e W be a non-equilibrium
point of the system (2), where A is the canonical malrix, B — iO, . , 0 , 1)*. Assume that a number ao satisfies the condition
(31)
• ( / V - i ( l ) 6 , 6 ) ' and T = 7'(jro, J:*) is a positive root of (6).
Then the conlrol (8), where zU) is the solution ofihe Cauchy problem (10), transfers the point XQ lo the point JC* in the lime T in global sense along the trajectory (9) and satisfies the preassigned constraints \uil)\ < dforall l e [0, T].
Pmof It is sufficient to show that the conlrol Uiz) of theform (15) satisfies the preassigned constrains \iiiz)\ < d for all z e R" if the number ao in i6) satisfies the condition i3l).
Using the equality
0e-^''-''BB*e-^~'^ - Di0)e-'^'BB^e''^''Di0).
where D ( 0 ) = diag (0^^'"^ I , we have the equalities
/V(0) = D ( 0 ) / V ( 1 ) D ( 0 ) . w - ' ( 0 ) = D - ' ( 0 ) A ' " ' ( l ) O - ' ( 0 ) . (32) P u t y { 0 ( z ) . ; ) = O"'(0(,-)),-. Po = - 5 f i * / V - ' ( l ) . Then, using (32), the control (15) takes Ehe form uiz) = 0 ~ 3 ( z ) p Q y ( 0 ( z ) , z ) . Consider the finding of extremum of the function 0 ~ 5 P o y ( 0 . z ) with restriction of the form ( A ' ~ ' ( l ) y ( 0 . z ) . i ( 0 . z)) - 2006^ — 0, where 0 is fixed. Let VQ be an extremal point Then Lagrange's method gives yo — y r 0 ~ l W ( l ) P Q . Pulling yo in the restriction, we obtain
--±^2ao/iNil)P^.P^)0.
Then. 0""^Po.v'o = ±y«o('V Hi)B. B)/2. Since ao satisfies the condition (31), then
|«(z)| < (/for all z eR". D
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V. I, Korobov, V. O, Skoryk Example 1 Consider the problem of construction of a restncted control, which transfers the point Xo — ix]o.X20,X2o)^ to the non-equilibrium point .v, — ix]],X2],X3i)' along the trajectory A-(;) of the system
-X] -h X2-1-X-} — u ], -2x] -I- 2 J : 2 -i- 2x-} -i- U] - K T .
The system (33) satisfies the conditions of Theorem I, moreover, rank(S, AB, A^B) = 3, and rank(fe|. Ab], A-b]) ^ 2, rank(&2. Ab2. A-fti) = 1. Hence, the system (33) is not completely controllable by a control U] or a control HI only.
The matrix W ~ ' ( 0 ) has the form
A'-'(0) = 2(6+5m 36+240+140' 6(6+40+30^) i2(l-t-t-J) 6(6+40+30^) 12(3+20+20^)
We choose ao from the condition (7) and we define the controllability function 0 ( s ) for all z / 0 from (II), which takes the form
2ao0'' - 80^zf - 2 ( l 8 -I- 120 + 7 0 ^ ) Zj - 12 (e -f 4 0 -j- 3 0 " ) Z2Z?, -36zl - 240zl - 240^zl -I- 4 0 z i ((6 + 50)z2 + 6(1 -f 0)z3) - 0. (34) The control uiz) from (15) has the form
30t;);i-3(2+0a))(zj+;i)\
0^(E) 1
where z = (si, Z2. Js)*-The control transfers an arbitrary point zo e M^ lo the origin in the time 0(zo) and satisfies the constraint ||H(Z)|| < 1. However, as T is unknown, then zo is also unknown, and hence the time is an object Eo be found Equation (6) takes the form
2aoT^-p]T^-p2T-p3^0, (36)
Pl - - 2 ( - 4^fo - IOA:f, - 7.v|u + 2X2OJ:21 - 13A:|, - 18;c2nx3o -I- 6:^21x30 - 12x3^0 -{-2x]oi-2xi] -^5^:20 + ^:21 +6x3o)-f 6(A:2O-5A:2I-|-2J:3O);C3| - 18;c|, +2.1:11 (JC2O+11J:21 + 12;C3I)),
P2 - - 2 4 ( . C , O + 2A:,, - A : 2 O - 2 J : 2 | - X30 - 2X^1) ix2o - X2] -i-X30 - X^]) , Pi = 36(A-2O -A:2I +^^30 -.^31)"
Let r be a positive root of (36), which has no less than one positive root. Then zo = (iio. £20.230)*. where zio ^ J:]O - (1 + T)x]] -]-T(x2] +JC31), Z20 = -27-^,1 -}-jc2o H fi Springer
Construction of Restricted Controls for a Non-ec|uilibrium Point
( 2 r - 1 ) « | + 27-13,, Z30 = Tx„ -TX2, + « 0 - ( l + r)j:31, and z(r) = (zi(r), Z2(r), Z3(t))*. where
ziC) = ^ ^ ( 2 ( n i o - r ( z 2 o + Z3o))cos7(/) + v^((2 + r - r ) r z i o - ( 6 - ((2 + 7-) + 7-(4 + r))(Z20 + Z30)) sin /(/)),
Z2(t) = ^ ^ ( j ' ( z i o - 2 z 2 o - 3 z 3 o ) - ( 7 - z i o + (2r-37")(Z20 + Z3o))cosy(t) +V2((7' - I - l)Tz,o + (3 - T - r ^ + r(2 + r))(z20 + Z30)) sm y ( r ) ) , Z3(I) = ^ ^ ( 2 r ( 2 z 2 o - z i o + 3z3o) + 2(Tzio + (/-27-)(z20 + Z3o))coS)-(t)
-Jl(TiT - , - 2)zio + (6 - 7-2 + 1(2 + T))(Z20 + Z3o)) sin / ( / ) ) , and yit) — V21n ^ , is the solution of the Cauchy problem (10). Then the control uil), which transfers the point zo to the origin, has the form
z i { ' ) - 2 z 2 ( / ) - 3 z 3 ( l )
Using (9), we obtain that (x,it) xit) = xiit)
^*3il)
'il + T-t)x„ - i r - t ) (.121+131)+ Z l ( l ) \
« ! + 2 ( 7 - r ) { A : , i - j : 2 i - J : 3 i ) + Z 2 ( t ) (38)
^ ( I - 7 ) ( j : i | - . V 2 i - i 3 t ) + .t3I+Z3(/) / is the trajectory along which the control uil) of the form (37) transfers the point .vo to the point X. in the time T.
Consider the case when .vo = (2, 3, — t)*, .v, = (i, 1, 1)'. We choose flo — 9/200 In this case, (36) takes the form ^T^(T- - 400) = 0, and T ^ TixQ. x.) = 20 is the unique positive solution of this equation. The control i((f) which transfers the point ;co to the point X. in the time 7 — 20 along the trajectory (38) of the form
I r - 1 9 + i ( I - 2 0 ) ( 2 ( t - 2 1 ) c o s y ( t ) - V 2 ( / - 2 4 ) s m / ( r ) ) >
xit) = 2( - 39 - 5^(r - 20) (3 + (39 - 2l) cos I'd) + V2(f - 18) sin 7(1))
^ 21 - t + i ( r - 20) (6 - 2(( - 1 9 ) c o s y ( 0 + ^/2(f - 16)siny(I)) , where yit) — V2\nil - 1/20) has the form
(cos (V2ln 22=^) + v ^ s i n (s/2in ^ ) ) \ (39)
uit) = I
Graphics of components of the control uil) from (40), the graphic of its norm and graph- ics of components of the trajectory xit) from (39) are represented in Figs. 1, 2. and 3.
t^spectively. Obviously, this control satisfies the preassigned constraints.
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V I. Korobov, V. O. Skoryk
F i g . l Graphics of components of the conlrol u{i)
l«(/)l
15 20
F i g . l Graphicof nonn of the control u(l)
Fig. 3 Graphicsofcomponenisof the trajectory J:(J)
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Construction of Restricted Controls for a Non-equilibrium Point
The results of Section 2 can be developed for arbitrary" linear systems and for nonlinear systems (1), which can be reduced to hnear systems.
i Ki:)robov, V.I * A general approach to the solution of the problem of synthesizing bounded controls in a control problem. Math. USSR Sb 37, 535-539 (1979)
2 Korobov, V.I Controllability Function Method. R&C Dynamics, Moscow-Izhevsk (2007) 3 Korobov, VL, Sklyar, GM • Methods of constructing positionjl controls and an admissible maximum
principle Differ Equ 26, 1422-1431 (1990)
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