By Jean-Michel Coron

This e-book provides how to research the controllability and the stabilization of nonlinear regulate platforms in finite and endless dimensions. The emphasis is wear particular phenomena because of nonlinearities. particularly, many examples are given the place nonlinearities develop into necessary to get controllability or stabilization. a variety of tools are offered to review the controllability or to build stabilizing suggestions legislation. the ability of those equipment is illustrated through a number of examples coming from such components as celestial mechanics, fluid mechanics, and quantum mechanics. The publication is addressed to graduate scholars in arithmetic or keep watch over thought, and to mathematicians or engineers with an curiosity in nonlinear keep watch over platforms ruled through usual or partial differential equations

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**Example text**

Evaluate f (15). How much glacier area disappears in 15 years? Solve f (t) = 12. 26. Graph the following function in the window −10 ≤ x ≤ 10, −10 ≤ y ≤ 10. Is this graph a line? Explain. y = −x x − 1000 900 27. Graph y = 2x + 400 using the window −10 ≤ x ≤ 10, −10 ≤ y ≤ 10. Describe what happens, and how you can fix it by using a better window. 28. Graph y = 200x + 4 using the window −10 ≤ x ≤ 10, −10 ≤ y ≤ 10. Describe what happens and how you can fix it by using a better window. 29. 23 shows the graph of y = x2 /1000 + 5 in the window −10 ≤ x ≤ 10, −10 ≤ y ≤ 10.

If m1 and m2 are not zero, we have the following result: Let l1 and l2 be two lines having slopes m1 and m2 , respectively. Then: • These lines are parallel if and only if m1 = m2 . 1 • These lines are perpendicular if and only if m1 = − . m2 In addition, any two horizontal lines are parallel and m1 = m2 = 0. Any two vertical lines are parallel and m1 and m2 are undefined. A horizontal line is perpendicular to a vertical line. 40. 41 shows l1 and l2 , two perpendicular lines with slope m1 and m2 .

D) Find a formula for the number of patrons as a function of profit. 50, how many patrons attended the theater? 41. The demand for gasoline can be modeled as a linear function of price. 10 per gallon, the quantity demanded in a fixed period is q = 65 gallons. 50 per gallon, the quantity demanded falls to 45 gallons in that period. (a) Find a formula for q in terms of p. (b) Explain the economic significance of the slope of your formula. (c) Explain the economic significance of the q-axis and p-axis intercepts.