How to change date to (Y-m-d)

how to change the date on start and end || the current format is m-d-Y

public function exportListContactAction()
{
  $criteria = $this->request->getQuery('criteria');
  $value = $this->request->getQuery('value');
  $period = $this->request->getQuery('period');
  $start = $this->request->getQuery('start');
  $end = $this->request->getQuery('end');

  $map = [
      "1" => "Name", 
      "2" => "Email",
      "3" => "Phone"];

      $criteria = $map[$criteria];

  $map1 = [
      "1" => "CreateDate",
      "2" => "CommentDate"];

      $period = $map1[$period];

      $siteId = $this->getSite();

      $options = $this->config->database;

      $dsn = "mysql:host={$options->host};dbname={$options->dbname};charset=utf8";
      $opt = [
          PDO::ATTR_ERRMODE            => PDO::ERRMODE_EXCEPTION,
          PDO::ATTR_DEFAULT_FETCH_MODE => PDO::FETCH_ASSOC,
          PDO::ATTR_EMULATE_PREPARES   => false,
          PDO::MYSQL_ATTR_USE_BUFFERED_QUERY => false
      ];

      $db = new PDO($dsn, $options->username, $options->password, $opt);
      $db->setAttribute(PDO::MYSQL_ATTR_USE_BUFFERED_QUERY, false);

      $db->query("SET TRANSACTION ISOLATION LEVEL READ UNCOMMITTED ;");

      $sql = "
      select c.*, r1.Description from Contact c 
      left join Reference r1 on c.TypeId=r1.Id 
      where c.SiteId=$siteId and c.ExpireDate='3000-01-01 00:00:00'";

      if ($criteria) {
          $sql .= " and c.`$criteria` like '%$value%' ";
      }

      if ($period) {
          $sql .= " and c.`$period` between '$start' and '$end'";
      }

      $sql .= " ";

      $query = $db->prepare($sql, array(      
          \PDO::ATTR_CURSOR => PDO::CURSOR_FWDONLY
      ));

      $db->query("COMMIT ;");

      $query->execute();