2900-458-R Memorandum of Project MICHIGAN AN OPERATOR THEORETIC FORMULATION OF LINEAR DIFFERENTIAL SYSTEMS W. A. PORTER April 1964 Navigation and Control Systems Laboratory #7"&&t4 4 e 4 ad 7 THE U N I V E R S I T Y OF MI C H I G A N Ann Arbor, Michigan

Institute of Science and Technology The University of Michigan NOTICES Sponsorship. The work reported herein was conducted by the Institute of Science and Technology for the U. S. Army Electronics Command under Project MICHIGAN, Contract DA-36-039 SC-78801; the U. S. Air Force under Contract AF-33(657)-11501; and the National Science Foundation under Contract GP-524. Contracts and grants to The University of Michigan for the support of sponsored research by the Institute of Science and Technology are administered through the Office of the Vice-President for Research. Note. The views expressed herein are those of Project MICHIGAN and have not been approved by the Department of the Army. Distribution. Initial distribution is indicated at the end of this document. Distribution control of Project MICHIGAN documents has been delegated by the U. S. Army Electronics Command to the office named below. Please address correspondence concerning distribution of reports to: Commanding Officer U. S. Army Liaison Group Project MICHGIAN The University of Michigan P. O. Box 618 Ann Arbor, Michigan DDC Availability. Qualified requesters may obtain copies of this document from: Defense Documentation Center Cameron Station Alexandria, Virginia Final Disposition. After this document has served its purpose, it may be destroyed. Please do not return it to the Institute of Science and Technology. ii

Institute of Science and Technology The University of Michigan PREFACE Project MICHIGAN is a continuing, long-range research and development program for advancing the Army's combat-surveillance and target-acquisition capabilities. The program is carried out by a full-time Institute of Science and Technology staff of specialists in physics, engineering, mathematics, and related fields, by members of the teaching faculty, by graduate students, and by members of other research groups and laboratories of The University of Michigan. The emphasis of the Project is upon research in imaging radar, MTI radar, infrared, radio location, image processing, and special investigations. Particular attention is given to all-weather, long-range, high-resolution sensory and location techniques. Project MICHIGAN was established by the U. S. Army Signal Corps at The University of Michigan in 1953 and has received continuing support from the U. S. Army. The Project constitutes a major portion of the diversified program of research conducted by the Institute of Science and Technology in order to make available to government and industry the resources of The University of Michigan and to broaden the educational opportunities for students in the scientific and engineering disciplines. Documents issued in this series of Technical Memorandums are published by the Institute of Science and Technology in order to disseminate scientific and engineering information as speedily and as widely as possible. The work reported may be incomplete, but it is considered to be useful, interesting, or suggestive enough to warrant this early publication. Any conclusions are tentative, of course. Also included in this series are reports of work in progress which will later be combined with other materials to form a more comprehensive contribution in the field. Progress and results described in reports are continually reassessed by Project MICHIGAN. Comments and suggestions from readers are invited. Robert L. Hess Director Project MICHIGAN 111

Institute of Science and Technology The University of Michigan ACKNOWLEDGMENTS The author gratefully acknowledges the exceptional teaching and research environment created jointly by the Department of Electrical Engineering and the Institute of Science and Technology of The University of Michigan. In particular, the help and encouragement of Electrical Engineering Prof. L. F. Kazda and Mr. James O'Day of the Navigation and Control Systems Laboratory, IST, have been indispensable in the prosecution of this research.

AN OPERATOR THEORETIC FORMULATION OF LINEAR DIFFERENTIAL SYSTEMS ABSTRACT In this report the linear differential system k(t) = A(t)x(t) + B(t)u(t) X(to)= x is reduced to a canonical operator theoretic form. This representation consists of a parameterized family of bounded linear transformations into a cartesian product of the underlying scalar field. It gives immediate results for the minimum energy control problem. INTRODUCTION A number of recent articles have utilized the function-space approach to the analysis of control problems. The attractiveness of this approach stems from the manner in which the technical "underbrush" is cleared away, leaving the essential features in clear view. In this report the linear time varying system differential equation 0 k(t) = A(t)x(t) + B(t)u(t) X(to) = x (1) is reduced to a simple operator-theoretical form which in itself yields considerable insight into the structure of the system. In Equation 1 the symbols x(t) and u(t) denote the column vectors x(t) = col(x1(t),..., xn(t)), u(t) = col(u1(t),..., um(t)), and A(t), B(t) denote n x n and n x m matrices, respectively. It is well known that under fairly general conditions the solution to Equation 1 exists, is unique, and is expressible in the following matrix integral form: xu(t, to, x) = b(t, t )x0 + b(t, t0) J (t, s)B(s)u(s)ds (2) U~~~ o (to'

Institute of Science and Technology The University of Michigan To make these statements precise, define T as a fixed interval on the real line; then the following theorem summarizes the properties important to this discussion [1, 2]. Theorem 1: Let IIA(t)II denote the norm of A(t) at time t; then if I A(t)l <_ m(t), where m(t) is an integrable function on T, there exists a unique matrix 4I(t, t ), which is absolutely continuous on T and satisfies the differential equation 4(t) = A(t)1~(t) c(to) = I t, t e T (3) 0 almost everywhere on T. In this theorem, it is assumed that I I A(t)| I is the norm induced by any suitable norm on x(t). The solution guaranteed by the theorem is, of course, the matrix 4(t, t ) used in Equation 2. 2 THE OPERATOR THEORETIC FORMULATION A moments reflection on Equation 2 emphasizes the extent to which the operations involved are concrete. It is also true that in many cases the abstraction of physical problems affords a clearer insight into the basic character of the problem. Such is the case with differential systems. To shorten the following treatment, let us agree on the standard notations: C(T) = {x(t) ] x(t) is a continuous function for t e T} L(T) ={x(t) x(t) is measurable1 on T and x(t)IPdt]1/P < 1 <p< A direct consequence of Theorem 1 is that if each element of B(t) is bounded and measurable on T, and if each u. E L1(T), j = 1..., m, then Equation 2 represents the unique solution of Equation 1. Theorem 1 states, in addition, that D(t, t ) is absolutely continuous, which implies that each element qoij(t) of this matrix is in C(T), i, j = 1,..., n. Since every continuous function takes on an absolute maximum and minimum on any finite interval, it is easy to show that yoij(t, to) e L(T) for any 1 < p oo and i, j = 1,..., n. Thus the functions qoij(t, to) may be considered as elements in any of these function spaces. That is, measurable in the sense of Lebesque. In the case p = co, the integral is replaced by ess. sup. Ix(t) i. teT 2

Institute of Science and Technology The University of Michigan It is not essential, but convenient, to assume that B(t) is also a matrix of continuous functions so that the matrix Z(t) = I(to, t)B(t) is a continuous n x m matrix. Let zij(t) denote the elements of Z(t) and let Zi(t) = [zil(t),..., Zim(t)] denote the i-th row of Z(t). Define the operation [zij, u](s) by m [zi, u](s) = Zij(s)uj(s) j=l Then the reader can easily verify that 4(to, s)B(s)u(s) is an nxl vector, the i-th component of which is given by [zi, u](s). Thus we have ~~~t t r[Zl' u](s) (t (t0, s)B(s)u(s) =Jds (4) o [ZnI u](s) In all cases, integrals of the form zij (s)uj(s) ds are involved. For fixed t, the form of this expression is that of a linear functional [3]. We have remarked earlier that if zij(t) is continuous on T, it qualifies as an element of any L (T) for 1 < p < oo. Thus. if u.(t) e L,(T), then z. (t) e L (T), where 1/p + 1/p' = 1, and the functional fii defined by 1J p t At fi (u) zij(s )u(s)ds 0 exists and is the general representation2 of an element of the space L* (T). P In many applications, the functions ul(t),..., um(t) represent independent inputs to a physical system. Hence it is desirable to consider them as elements from distinct function spaces. Restricting ourselves to Lp (T) spaces, we have P uj(t) e L (T) j = 1,...,m 3 Let the space U be defined by the equation. (T) XLp (T) X... XL (T) P 1 Pm 2If X is a linear space, then X* denotes the space of all bounded linear functionals defined on X. X* is called the conjugate of X. See Reference 3, page 185, for details. 3This notation denotes the cartesian product of function spaces. See Section 4, page 121 of Reference 4. 3

Institute of Science and Technology The University of Michigan Then every input vector u(t) = (ul(t),..., um(t)) is an element of U. The functionals fit i = 1,..., n, on U can be defined in the natural way by m m t t f. (U) fi (U) = Zij (s)uj(s)ds [zit uZ ](s)ds j=1 j=1 o o where, indeed f. e U* = L* (T) x. X L* (T) i = 1,..., n t p' P' 1 m In terms of these definitions, we can now reformulate Equation 2. First, using Equation 4, we note that the forced response can be rewritten as t pKit -(t, t0),t 4(to, s)B(s)u(s)ds = b(t, t) ft(u (5) Let each functional f. (u) be written in the dyadic notation f.t(u) <fi X u>. Then, denoting the 1 1 1 j-th column of ib(t, to) by oj(t, to)>, we see that Equation 5 reduces to t n C (t t (p (t s)B(s)u(s)uds ) d=i(t, t )> <fi,u > 0 i=1 If the operator F is defined by F j- =1 i(t' to)> fi j=1 then we have just proved the following theorem. Theorem 2: Every differential system obeying x(t) = A(t)x(t) + B(t)u(t) x(t) = 0 (6) for which f I IA(t) 1i dt < oo and u E [L1(T)]m can be described as a parameterized family of 4 t bounded linear transforms F each with finite dimensional range. 4The proof that Ft is bounded follows from the fact that each f. is bounded. 4

Institute of Science and Technology The University of Michigan In the terminology of Theorem 2, the variable t is considered as parameter of the transformation. The columns o1(t),..., On (t) of the transform matrix span the range space of t t F for every value of the parameter t. The range R of F is then contained in L(ml(t),..., "(t)) = R. Ft is then a parameterized mapping of U into Rn, written F:U -Rn. Let us consider the initial-condition response of the system. This term is given by b(t, t )XO If Rn denotes the space of real n-tuplets, then clearly x E R. If e,.., en den 0 0 notes the rows of the identity matrix on Rn, it is also clear that the i-th component x. of x~ is given by <el, x > and, hence n?(t, t )x~ = pi(t, t0)><ei, x > i=l Let us define the linear transformation Jt by $t,_ E si(t' to )> < ei (7) i=1 Then 4(t, to)xo~ xt andJt: Rn -L(o1(t),...., On(t). Observe also that the ranges of Jt and Ft are identical. J t however, is one-to-one and onto and, hence, is nonsingular. It is possible to incorporate the total system response within the present framework. To do this, let the augmented input space V be defined by V = Rn x U. The elements of V are then of the form v = [x1.., xn, u1(t),..., u (t)]. Noting that Equation 2 has the form Xu(t, to, x) J x + F u, let us define the operator Tt = Jt Ft by Ttv =Jx + F u (8) Since J and F have the same range space, namely Rn = L(4o1(t),., S(t)), it is clear that t t t T:V - L(pl(t),.',p(t)) and that T is linear. From this it follows that T must have an n-term dyadic expansion. In fact, let the functionals {g1..., gn} on V be defined by git(v) = < ei, x~ +<fit, u> Then, combining Equations 6, 7, and 8, we have n n n xu(t, t; x) = T v = Oi t0)><ei, x > + Oi(t t0)><f., u> = qi(t, to)><git, v> (9) i=l i=l i=l

Institute of Science and Technology The University of Michigan The following theorem summarizes these results. Theorem 3: If A(t) is a square matrix for which fT I]A(t) | dt < cc, and if each element of the vector B(t)u(t) is integrable on T, then the differential system x(t) = A(t)x(t) + B(t)u(t) x(t) = x~ t, t e T can be represented as a parameterized family of bounded linear transformations onto a finite dimensional range. 3 DISCUSSION The intent of this report is not the detailed exploitation of the above representation, for the implications are many and varied. It seems appropriate, however, to mention in passing some of the salient advantages of this approach to systems analysis. First let us note that generalization of the present result to systems of the form x:(t) = A(t)x(t) + B(t)u(t) x(t) = x y(t) = C(t)x(t) + D(t)u(t) can be managed with little modification in the previous arguments. More important is the fact that although this report isconcerned with continuous time systems, it can be shown that discrete time and, in fact, any linear dynamic system can be reduced to a representation of the above form [5, 6]. Thus the present approach emphasizes the similarities rather than the differences between the various types of linear systems. The form of Equation 5 itself is worth comment. Let us inquire into the linear dependence t t or independence of the functionals <fit,..., <f. Definition: Any operator A of the form n ~ ~ 1'.n A = 4j.><f. will be called P-normal if the sets lo>'.'.,n> and <fl''''' <f are both j=l linearly independent. In the present case, the ol,..., p are the n-linearly independent solutions to the homogeneous differential system and the independence of the set F = {<fl,.. <n is the only issue. 6

Institute of Science and Technology The University of Michigan The definition of P-normality has a close relationship to the oft-mentioned general position condition [7] and the practically identical controllability concept [8]. It can be shown that the P-normal condition is both more general and satisfying in that it covers discrete, continuous, and composite systems in a single stroke [9]. Let the input spaces be restricted to being replicas of the Hilbert space H = L2(T). The space U is also a Hilbert space (with respect to the usual inner product for cartesian spaces) and the functionals <f. on U are inner products with vectors in U. Let us denote the functional and the vector by the same symbol fj, j =1,..., n. Then, if L(fl,...,fn) is the linear manifold spanned by fl,'.. fn we may make the decomposition U = L E) L, and from Equation 5 it is clear that the null space NT of T is equal to L. Since L is n-dimensional, this implies that there are only n-linearly independent input signals which "efficiently" affect the output. Reference 10 presents a thorough discussion of this and other matters in a more general setting. Finally, let us note that the operators J and Tt, defined by Equations 5 and 8, are time varying in nature. If a fixed-arrival-time problem is formulated in the present manner with arrival time t = tf then T:V - Rn. In this case any basis for Rn can be used to decompose T I, since the relation between the various decompositions is simply a nonsingular change of variables. If the arrival time tf is not fixed, but is to be determined (as in the Bolza problem of the calculus of variations), tf may be considered as a parameter; this case is treated as before. RE FERENCES 1. W. A. Porter, Modern Foundations of Systems Analysis, Textbook, to appear, Chapters I and III. 2. E. A. Coddington and N. Levison, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955, Chapter II. 3. A. E. Taylor, Introduction to Functional Analysis, John Wiley, New York, 1958, p. 33. 4. G. F. Simmons, Introduction to Topology in Modern Analysis, McGraw-Hill, New York, 1962, p. 121. 5. W. A. Porter, Representation Problems in General Dynamic Systems, Report No. 2900-463-R, Institute of Science and Technology, The University of Michigan, Ann Arbor, Mich., in publication. 7

Institute of Science and Technology The University of Michigan 6. W. A. Porter, A Functional Decomposition for Linear Discrete Systems, Report No. 2900-459-R, Institute of Science and Technology, The University of Michigan, Ann Arbor, Mich., in publication. 7. L. S. Pontryagin et al., Mathematical Theory of Optimal Processes, John Wiley, New York, 1962. 8. R. E. Kalman, Y. C. Ho, and K. S. Narendra, Controllability of Linear Dynamic Systems Contributions to Differential Equations, Vol. I, John Wiley, New York, 1961. 9. W. A. Porter, "The P-Normal Condition for the General Linear Dynamic System," Article, to appear. 10. W. A. Porter, "A New Approach to the General Minimum Energy Problem," Proceedings of the Joint Automatic Control Conference, Summer 1964, to appear. 8

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+ + -F ~~~~~+ AD Div. 19/4 UNCLASSIFIED AD Div. 19/4 UNCLASSIFIED Inst. of Science and Technology, U. of Mich., AnnArbor I. Title: Project MICHI- Inst. of Science and Technology, U. of Mich., AnnArbor I. Title: Project MICHIAN OPERATOR THEORETIC FORMULATION OF GAN AN OPERATOR THEORETIC FORMULATION OF GAN LINEAR DIFFERENTIAL SYSTEMS by W. A. Porter. II. Porter, W. A. LINEAR DIFFERENTIAL SYSTEMS by W. A. Porter. II. Porter, W.A. Memo. of Project MICHIGAN. Apr. 64. 8 p. 10 refs. III. U.S. Army Electronics Memo. of Project MICHIGAN. Apr. 64. 8 p. 10 refs. III. U.S. ArmyElectronics (Memorandum No. 2900-458-R) Command (Memorandum No. 2900-458-R) Command (Contract DA-36-039 SC-78801) IV. Contract DA-36-039 (Contract DA-36-039 SC-78801) IV. Contract DA-36-039 (Project No. 3D5801001) Unclassified memo. SC-78801 (Project No. 3D5801001) Unclassified memo. SC-78801 In this report the linear differential system V. Project No. 3D5801001 In this report the linear differential system V. Project No. 3D5801001 VI. U. S. Air Force VI. U. S. Air Force i(t) = A(t)x (t) + B(t)u(t) x(to) =x VII. Contract AF-33(657)- i(t) = A(t)x (t) + B(t)u(t) x(t)= VI Contrac 11501 11501 is reduced to a canonical operator theoretic form. This VIII. National Science Founda- is reduced to a canonical operator theoretic form. This VIII. National Science Foundarepresentation consists of a parameterized family of tion representation consists of a parameterized family of tion bounded linear transformations into a cartesian product IX. Contract GP-524 bounded linear transformations into a cartesian product IX. Contract GP-524 of the underlying scalar field. It gives immediate re- of the underlying scalar field. It gives immediate results for the minimum energy control problem. sults for the minimum energy control problem. Defense Defense (over) Documentation Center Documentation Center (over) UNCLASSIFIED (over) UNCLASSIFIED + + + AD Div. 19/4 UNCLASSIFIED AD Div. 19/4 UNCLASSIFIED Inst. of Science and Technology, U. of Mich., Ann Arbor I. Title: Project MICHI- Inst. of Science and Technology, U. of Mich., Ann Arbor I. Title: Project MICHIAN OPERATOR THEORETIC FORMULATION OF GAN AN OPERATOR THEORETIC FORMULATION OF GAN LINEAR DIFFERENTIAL SYSTEMS by W. A. Porter. II. Porter, W. A. LINEAR DIFFERENTIAL SYSTEMS by W. A. Porter. II. Porter, W. A. Memo. of Project MICHIGAN. Apr. 64. 8 p. 10 refs. III. U. S. Army Electronics Memo. of Project MICHIGAN. Apr. 64. 8 p. 10 refs. III. U. S. Army Electronics (Memorandum No. 2900-458-R) Command (Memorandum No. 2900-458-R) Command (Contract DA-36-039 SC-78801) IV. Contract DA-36-039 (Contract DA-36-039 SC-78801) IV. Contract DA-36-039 (Project No. 3D5801001) Unclassified memo. SC-78801 (Project No. 3D5801001) Unclassified memo. SC-78801 In this report the linear differential system VI US. Air Forcet In this report te linear dI. UD. Air Force 0 ~~VI. U. S. Air Force 0VI. U. S. ArFoe i(t) =A(t) x (t) + B(t)u(t) X(to)= x VII. Contract AF-33(657)- i(t)= A(t) x (t) + B(t)u(t) x(to) = x VII. Contract AF-33(657)11501 11501 is reduced to a canonical operator theoretic form. This VIII. National Science Founda- is reduced to a canonical operator theoretic form. This VIII. National Science Foundarepresentation consists of a parameterized family of tion representation consists of a parameterized family of tion bounded linear transformations into a cartesian product IX. Contract GP-524 bounded linear transformations into a cartesian product IX. Contract GP-524 of the underlying scalar field. It gives immediate re- of the underlying scalar field. It gives immediate results for the minimum energy control problem. sults for the minimum energy control problem. Defense Defense (over) Documentation Center Documentation Center (over) UNCLASSIFIED (over) UNCLASSIFIED + + +

AD UNCLASSIFIED AD UNCLASSIFIED DESCRIPTORS DESCRIPTORS Control systems Control systems Differential equations Differential equations Navigation Navigation Theory Theory Topology Topology UNCLASSIFIED UNCLASSIFIED + AD UNCLASSIFIED AD UNCLASSIFIED DESCRIPTORS DESCRIPTORS Control systems Control systems Differential equations Differential equations Navigation Navigation Theory Theory Topology Topolog UNCLASSIFIED UNCLASSIFIED

+ + AD Div. 19/4 UNCLASSIFIED AD Div. 19/4 UNCLASSIFIED Inst. of Science and Technology, U. of Mich., Ann Arbor I. Title: Project MICHI- Inst. of Science and Technology, U. of Mich., Ann Arbor I. Title: Project MICHIAN OPERATOR THEORETIC FORMULATION OF GAN AN OPERATOR THEORETIC FORMULATION OF GAN LINEAR DIFFERENTIAL SYSTEMS by W. A. Porter. II. Porter, W. A. LINEAR DIFFERENTIAL SYSTEMS by W. A. Porter. II. Porter, W. A. Memo. of Project MICHIGAN. Apr. 64. 8 p. 10 refs. III. U. S. Army Electronics Memo. of Project MICHIGAN. Apr. 64. 8 p. 10 refs. III. U. S. Army Electronics (Memorandum No. 2900-458-R) Command (Memorandum No. 2900-458-R) Command (Contract DA-36-039 SC-78801) IV. Contract DA-36-039 (Contract DA-36-039 SC-78801) IV. Contract DA-36-039 (Project No. 3D5801001) Unclassified memo. SC-78801 (Project No. 3D5801001) Unclassified memo. SC-78801 In this report the linear differential system VI U. S. Air Force In this report the linear differential system VI U. S. Air Force VI. U. S. Air Force 0VI. U. S. Air o i(t) = A(t) x (t) + B(t)u(t) x(to) =x VII. Contract AF-33(657)- k(t) = A(t) x (t) + B(t)u(t) X(to) = VII. Contract AF-33(657)11501 11501 is reduced to a canonical operator theoretic form. This VIII. National Science Founda- is reduced to a canonical operator theoretic form. This VIII. National Science Foundarepresentation consists of a parameterized family of tion representation consists of a parameterized family of tion bounded linear transformations into a cartesian product IX. Contract GP-524 bounded linear transformations into a cartesian product IX. Contract GP-524 of the underlying scalar field. It gives immediate re- of the underlying scalar field. It gives immediate results for the minimum energy control problem. sults for the minimum energy control problem. Defense Defense (over) Documentation Center (over) Documentation Center (over) UNCLASSIFIED UNCLASSIFIED + + AD Div. 19/4 UNCLASSIFIED AD Div. 19/4 UNCLASSIFIED Inst. of Science and Technology, U. of Mich., AnnArbor I. Title: Project MICHI- Inst. of Science and Technology, U. of Mich., Ann Arbor I. Title: Project MICHIAN OPERATOR THEORETIC FORMULATION OF GAN AN OPERATOR THEORETIC FORMULATION OF GAN LINEAR DIFFERENTIAL SYSTEMS by W. A. Porter. II. Porter, W. A. LINEAR DIFFERENTIAL SYSTEMS by W. A. Porter. II. Porter, W. A. Memo. of Project MICHIGAN. Apr. 64. 8 p. 10 refs. III. U. S. Army Electronics Memo. of Project MICHIGAN. Apr. 64. 8 p. 10 refs. III. U. S. Army Electronics (Memorandum No. 2900-458-R) Command (Memorandum No. 2900-458-R) Command (Contract DA-36-039 SC-78801) IV. Contract DA-36-039 (Contract DA-36-039 SC-78801) IV. Contract DA-36-039 (Project No. 3D5801001) Unclassified memo. SC-78801 (Project No. 3D5801001) Unclassified memo. SC-78801 In this report the linear differential system V. Project No. 3D5801001 In this report the linear differential system V. Project No. 3D5801001 VI. U.S. Air Force VI. U.S. Air Force i(t) = A(t) x (t) + B(t)u(t) x(t) = VII. Contract AF-33(657)- ki(t)= A(t) x (t) + B(t)u(t) x(t) = x VII. Contract AF-33(657)11501 11501 is reduced to a canonical operator theoretic form. This VIII. National Science Founda- is reduced to a canonical operator theoretic form. This VIII. National Science Foundarepresentation consists of a parameterized family of tion representation consists of a parameterized family of tion bounded linear transformations into a cartesian product IX. Contract GP-524 bounded linear transformations into a cartesian product IX. Contract GP-524 of the underlying scalar field. It gives immediate re- of the underlying scalar field. It gives immediate results for the minimum energy control problem. sults for the minimum energy control problem. Defense Defense (over) Documentation Center Documentation Center (over) | UNCLASSIFIED (over) UNCLASSIFIED + +

AD UNCLASSIFIED AD UNCLASSIFIED DESCRIPTORS DESCRIPTORS Control systems Control systems Differential equations Differential equations Navigation Navigation Theory Theory Topology Topology O -...., UNCLASSIFIED UNCLASSIFIED ---- AD UNCLASSIFIED AD UNCLASSIFIE D DE SCRIP TORS DE SCRIP TORS Control systems Control systems Differential equations Differential equations Navigation Navigation Theory Theory Topology Topology UNC LASSIFIE D UNC LASSIFIE D